(SANITIZED)UNCLASSIFIED PAPERS BY SOVIET BLOC SCIENTISTS ON MATHEMATICAL STATISTICS AND PROBABILITY(SANITIZED)

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80T00246A011700340001-4
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 362
• Document Creation Date: December 22, 2016
• Document Release Date: July 9, 2009
• Sequence Number: 1
• Publication Date: November 17, 1960
• Content Type: REPORT

Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
Next 1 Page(s) In Document Denied
Q
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 ~ ~
.~ ~
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
TEOP~iSi O~i0P0,uHbiX CJIY~IA1~Ii1~ TI07IE~
A.LI.Arnota (i;~ocxBa, AxaAeMHa xayK CCCP)
(~i-1. l~u. 1 `~u-~t Ly ~~ ~~ e~~>~ c~ ~ ~ ,~- h ~`"" `~ ~"v`' ~t ,e x`.c. L
I. BBeAeHAe
L~eHTpanbHbIM (~aKTOM xoppennuuoxxo~ TeopHH cTauHOxapHbix cny-
~la~HhlX npOueCCOB ABIIA@TCA (~aKT CyIgeCTBOBaHYIA CneKT 8nbH0T0 ~8,3-
no~ xax AnA ca~oro npouecca ~ ~t~
(I? I) ~ (t) = J e tai Cdr
T8K Yi AnA er0 xoppe~nauHOxxor~ c~yHKuYI1R .~~`C~
Ilpvi 3T0~ B cny~ae oAHOpoAHOro none ~ ~t~ , t E R.n, n~6xe
AononxLiTenbH~e npeAnono~cexxA o ero c1QbfMeTpHH ~oryT 6biTb cBeAeH~
K cneuHanbH~a orpaHH~eHUA~ Ha xoppenAuHOxxy>e c~yxxux~ .1~~'C ~ H
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
3Aecb ~ ~~~ - aAAHTHBHaa cny~a~HaA c~yxxuHa mHO~ecTBa (cny~a~Haa
(cu. HYI~{e c~opa~yn~t (2.2I) - (2.23) ).
net Ha. npOY[3BOJIbHO~ noKanbHO KOUnaxTHOI~ KoMMyTaTPlBHO33 rpynQe
npocTpaxcTBe ~.ri) x Ana eke 6onee o6~ero Knacca oAHOpoAb~x nod
(T. e . AnA o,uHOpoAHnx cny~ar~xux none ~ ~t~ ~ B n.-~eepHOm
xapH~ax npoueccoB c aeHOro~eepHnza Bpe~eHH~ napaweTpoaat=(t,,t;,...,tw
-~ x + ~
AHanorvc~Hne cnexTpanbHne pasno~eHHa cyr~ecTBy>oT AnA cTauxo-
pa~eTpo~ npeAenn HHTerpHpoBaHxa B (I.I) H (I.2) HaRo aaueHHTb xa
Be~tecTBeHH~ae sHa~eHHR ; AnA cny~airiH~x npoueccoB c AHCxpeTH~a na-
~epa), a 3"(~) - o6n~Haa HeorpHuaTenbxaa orpaxH~eHHaa raepa xa
(-~, ~) cBaaaHHaA c .~(1~) cooTHOmexHem
Bpeasexxo~i napaa~eTp t npouecca ~~ npeAnonaraea~ npxHr~a~o~x~e Hce
 6"!'~s~ Wit`., ;'.'"', ~ ~ ~'-*'~? ~~.~~ ~.,~
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
- 2 -
cnexTpanbxy>o ~epy ~'(~) Cana ~ ~n~ ). Oco6exHO HHTepecH~e AnA
npuno~exu~ ABnaeTCA cny~a~ oAHOpoAHOro u u3oTponxoro cny~ai~HOro
nonA - oAHOpoAHOro nonA ~ (t~ , o6naAa~u~ero c~epu~ecxo~l cuaa~eeT-
pue~. Odt~ui~ BHA KoppenAl(LIOHHO~J OGyHKuYlu .~(`~) ; `C:I~I , TaxOrO IIOAA
~aeTCa uaBecTHO~ c~op~ayno~i llleH6epra ~ I~
~ ~ ~~~~
c (~.~) s
rAe ~n-z - (~yHKIjMA beccena nopAAKa n2 a ~(a~ - orpaxxttea-
2
HaA Hey6bIBaI0I1(aA C~,yHT4I~LIA Ha ~0~ ~~ ). B CTBTLiCTU~eCKQ~! TeOpI~lH Typ- '
6yneHTHOCTYI paCClaiaTp7rIBa~1TCA TaK]rie BexTOpH~e oAHOpoAHne u u3oTpon-
Hne cny~a~Hxe nonA ~(t~-{~s~t~~...,~~,~t)}; OTBe~a~ou>,aA u~ xoppen~auHOHxaa
~aTpHua .~~~(~~=E{~~(t+ti')~~(t)~Hr~eeT cave "cnexTpanbxoe pa3no~exue",
Hecxonbxo Hanoa~uxa~u~ee (I.4) (c~a. ~2 - 4~ ). 06o6~gexue~~noHATyiA
B6KTOpHOrO LI30TpOnHOrO cny~a~HOro nOnA BBJIAeTCfi nOHATUe u30Tp0II-
H02'0~ Cny~Ia~IHOrO nOTOKa, BBeAeHHOe Y~TO ~5, ~ AnA TaKYiX nOTOROH '
TBK~{e CyIgeCTByeT CneKTpanbHOe pa3n0~{eHYie CneljYlanbHOrO BLIAB..
HeCKOnbKO B CTOpoHe OT pa60T n0 TeOpux cny~a~Hxx nonei~ H
eBKnLIAOB~iX npOCTpaHCTBBX CTOMT He60nbmaA 3aMeTKa A.M.06yXOBa ~6~ ,;~
nocBa~gexxaa oAHOpoAH~ cny~a3~Hn~ nonAm Ha crepe rJ2 TpexMepxoro p
npocTpaHCTBa R,3 . 06yxoB pacc~oTpen pasnozxeHUe Taxoro tionA_.~(A,`~~
B pAA no cc~epu~ecxur~ ~yHxuua~a y~ (8, ~j
(I.5} ~ (9, `P~ _ ~ ~- hem y'" (9, ~,
er o m=-c e
u noxaaan,~TO ycnoBUe oAHOpoAHOCTU none paBHOCUnbxo ycnoBx~
(I.6) E~em ~eifii . SCe~~h-ms?~e
B cony Teope~ax cno~eHUa AnA npucoeAuxeHHxx c~yaxuu~ ne~aHApa or-
c?Aa AnA xoppenAuuoxxo~f c~yxKuuu ,~j(Ai2~=E ~(a,,~s)~(ez~i)
(rAe 8,Z - yrnoBOe paccxvH~xe n~e~~+ To~IKa~au tai, ~i~ u (A~,Y~)
cc~epx SZ) nony~aeTCa npe~~TaBneHxe
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
- ~ .. -
?? Ze+s
(I.?) ,'~ CB~ _ ~, ~ ~ Cos 9~ , ~e = 2 ~~' 07
Q-o 2
Ha~AeHHOe B ApyrO~ CBA3H en~e paxbme Wex6epro~[7j 06paTao,npK
H ~e yAoaneTeopA~utHx (I.6) H Taxisx, PTO pAA (I.7) cxo-
n~6mx ~e
M
;,
AMTCA, none (I.5) i~y~eT oAaopoAx~ne, a e~yxxuHA (I.?) 6yAeT xoppe- ~ ~~~
nAuYI0HH0~ C~yHKI(Ylel~ OAHOpOAHOrO CnygaYlHOrO n01IA.
PesynbTaT~a (I.2), (I.4) H (I.7) HnH (I.I) H (I.5). BHemxe
npeAcTaBnAwTCA Eecb~aa pa3nH~x~,iH.EcTecTBebao ,uy~aTb,oAaaxo, uTO.
Bce oxH Aonsx~ coAepxcaTbcA B xa~ecTBe ~acTx~x cny~aeB B xaxo~-To
o6ueif TeopHH, oxaaTnBa~~e~ wHpoxHe xnacc~a cny~a~xbtx c~yaxuH~, xx-
.BapHaxTxnx oTxocHTenbxo xexoTOp~ax rpynn npeodpasoBanH~. HeTpyAao
noHATb, ~ITO e~,xHCTBBHHbIEd M8Te2daTH~eCKI~ annapaTOA~I, npHrOAHbiM AnA
-
-
-
..
4~~'
i1VVilJVNa~uA awaavu vvayv?? ~....~.....7
(2.320 ~(~~ _ f ~T'?[a) ~ d
yj d ~~~ ~ ~ ~ a~ ,
9 ~~~,, ,;,,~.~. _
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
^ ws^^ v^ ^ ^arns r^. uvvr .e.+'^.b. ^
- I5 -
rAe B cny~ae none~i, oAHOpoAxbix oTHOCUTenbxo ~?e~~x cABHroB ,
(2.~3') E~cd(~1 ~) ~~e(~~) =sde3~K(ninn2), ~ .
0.g C11yYzQe ^oaHOp*yoaNO/C~ru ojHOCU2jes6N/O, npa/6~.~x caBuze~
Cz.3y', E~iil (1,1) ^'KQ~"z~ = sK J~Q (Lli ~1.C,2).
~JIA noJIeu, OAHOpOAH~ OTHOCYITCnbHO RByCTOpoHHYIX cABHroB,.
AOJI~HbI Bb1Ti07IHATbCA H COOTHOIIIeHLlA (2.33) YI COOTHOlII2HYIA (2.34).
~TC~OAa B~ITexaeT, PTO KOppeltAuLiOHHaA c~yHKL(LIA -~~~~ 3AeTb AOJI~HB
~~ ~
pa3naraTbcA no cneAa~a onepaTOpoB npeAcTaBneHxs T ~~ ~ .Tax xax
6ecxoxe~HO~aepHxe yHUTapH~e onepaTOpn He r~e~T xoHe~Hxx cneAoe,
To oTC~Aa cne~yeT,uTO HexordnaxTHxx Hexo~a~yTaTHBH~x rpynnax, Bo-
o6ge roBOpa, ~o~eT H xe cyutecTBOBaTb cny~a~Hbtx none~i,oAxopoAHnx
n
oTHOCHTenbxo ABycTOpoxxHx cABHroB (cu.,Bnpo~e~,xr~~e cTp. ).
B saxn~o~exHe npHBeAeia xexoTOpne ccxnxH xa~pa6oTx, coAep$a-
(a)
qHe aBxne c~op~ynn AnA onepaTOpoB npeAcTaBnexx~ ~T (~) xa Tex
HnH HH~x rpynnax ; HcxoAa us 3Txx c~op~ayn, oueBHAxo,~ ~o:~HO sBHo
Ilepe~ILIC1lYITb TaK~ e H BCe ORHOpOAH~Ie n01IA Ha COOTBBTCTBy~IgMx rpylI-
nax. Ana rpynnn BpageHH~ Tpex~aepx+oro aBxnHAosa npocTpaxcTBa aHa-
~eHlli.'i BCL'X ~daTpYl~IHMX 3AeMeHTOB lil~,~QQ) (B ~yHxuYlAX OT TpeX
~ 0 i
yrnoB 3r~nepa, oAHOaxa~xo sa~aegRx BpageHHe~ ~ ) BxnxcaH~ B xxH-~
re ~23~ ; B o6ge~ cny~ae 3TYI MaTpYI~Hbte 31IemeHTb1 B~tpaata~TCA ~e-
pes .TpKroxo~aeTpHUeexHe c~yaxuvrH H nonvcHO~n fixo6H. Ana rpynnn ABx-
~eHU~ eBxnKAoaoH nnOCKOCTH (r~leIDge~! OfiIiOnapan~eTpHUeCKOe Ce1~e~fiCT-
Bo 6ecxoxe~HOraepHxx yHHTapxwx npeAcTaBneHU~i) ABHni1 BHA a~aTpx~xbix
3neraeHTOS T`~'~ ) yxasaH B pa6oTe ~24] (3Tli ~aaTpx~H~ae ane~ea-
d
TM Bbtpa~a~TCA ~epe3 TpHroHOUeTpH~ecxHe H 6ecceneBn c~yxxuxx).
Ana rpynnx ABxz~eHHi3 nnocxocTH JIo6a~eacxoro ~aaTpx~H~e a~ce~aeHT~
T`'~'(~) BwnvtCaHbt B p800Te ~25, - 3Aecb OHx B>apa~a~TCa ~epea
rKnepreo~aeTpK~ecxHe c~yxxuHH. B o6geM cny~ae npoH3BOnbxo~ xnaccra-
~ecxo~ rpynnn aBH~e Bxpa~eHHa Ana onepaTOpoB ~'~'~~~~~ (3aAaBae-
Mbtx He B aaaTpH~HOn ~opaae) ~ao~cHO Ha~Ti~ B xxHre ~26~ McTOAx no-
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
cTpoexHn c~opMyn Ana Bcex ~aTpH~xnx.aneMexTOB npeAcTaanexN~ rpyn;a
n~ Bpa~exN~ N rpynn~ ABN~:exNii eBxnH,~oBa n.-Mepxoro npacTpaacTea
v: rpynn>~ ABNx~:exN~ tt.-Mepxoro npocTpaxcTBa JIo6ayeBCxoro yxaaaH~r e
saMeTxax ~2?~ , ~28~ N ~29~ (3araeTNr~,~TO 3TLI ~aaTpN~xne 3nea~eHrg
B~tpa~:a~TCA ~epeo xexoTOpxe HoBne,Ao cHx nop xe Hayuexa~e, Tpaxc-
uexAexTxne ~yxxuxx). ~~
3. O~xopoAxne none xa oAHOpoAxMx npocTpaHCTBax.-
TlycTb X={x} - npoHaBOnbxoe oAxopoAxoe npocTpaacTao,T.e:
npocTpaxcTBO, AonycKarou~ee TpaH3HTHBxy>c rpynny npeo6pasoBaB~~~~: ~:
~={~} 06osxa~Hra ~epe3 ~={k,} ~cTauNOxapHy~ noArpynny ~ ,r.e.
noArpynny ocTaanA~~y ~ xa MecTe.xexoTOpy~ To~xy xoE X . CoBOSyn-
HOCTb npeo6pa3oaaHNir ~~ E ~ , nepeBOAar~vtx xo a c~NxcNpo$aaey~
TO~Ky xs F X , 06pa3yeT, O~IeBYi~HO, neBx~ KnaCC C1~e~HOCTYI ~1.~
rpynnx ~ no ~ TeM camera ~ae~c~y To~xaMH x, E X H nea~u xnac-
camH cMe~xocTH ~,~ ycTaxaBnraeaeTCA BaaNmxo oAxoaHagaoe coo~aeT-
CTBLie, T8K ~ITO X 1~OMH0 OTO~CAeCTBNTb C mHO}KeCTB01~1 3TNX KnaCCOH
Cme~{HOCTYl:. .!~' ~/~ ~ eCnM x1= ~~, 7o Qxl = ppi~' B AaAbHe~-
d
IDem mx BcerAa 6yAem npeAnonaraTb,~TO apynna y TononorN~ecsaA,
~ - ee 3amxHyTaa xomnaxTHaa noArpynna. B ~acTxom cnyuae,xor-
Aa ~ - eAHxH~xaA noArpynna, e~xopo,~xoe npocTpaHCTBO X o6pa~ga-
eTCa B rpynnoaoe npocTpaxcTBO ~={~}.
~TononorHa rpynno$oro npocTpaxcTBa ~ ecTecTeexxo Nx~-~gxpyer
TononorH~ B X (cm. ~IO~ ). IIpN 3TOm c~yxxuv~x xa X oxa3~Ba>orcA
xenpepxBxnMH TorAa N TonbKO TorAa, xor~a xenpepmBHx cooTBerc~-
By~~He c~yxxuNN xa ~. (npHxxma~~Ne nocTOAxxne 3Ha~eHNA Ha Bcex
neBMx xnaccax cmexcxocTx ~ nod ). Cny~a~HOe none fi(x) Ha X
mx onpeAenHm, xax senpep~Bxoe oTO6pa~exxe X B ~. . IIone fi(x)
xa3blBaeTCA OAHOpOAHbU9, ecnH ero nepB~ce H BTOpb1e 1~OmeHTbi He Me-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
- e.. ..^ r rvcaanr. WYt},~ik. 68'?EiL.1
- I7 -
HA~TCA npH npeoGpa3oaaHHAx ~ E ~ , T.e. ecnv[ E~(x~ pasxo no-
CTOAHHO~f (KOTOpyIO Mm Bcer,~a 6yAeM :CUHTaTb paBHO~i Hyn~), a
.~(x,,xz)=E~(x1,.~(x2~ _y~,OBneTBOpfleT co:oTHOmeHx?:
(3. I) ,13(x,,zz~ _ .1~(~x1,~xZ), ~ E ~.
OueBHAxo, uTO xnacc oAHOpoAHbtx cnyua~iHbix none~c Ha ~ , coHnaAaeT
C K1Ia000M O~HOpOAHhIX Cnyu0~Hb1X nO1Ie~f Ha :~. , noCTOAHHbUC .H8 BC6X
neBnx xnaccax CMe~HOCTH n0
IiauxeM onATb c npocTe~wero cnyuaA, xorAa rpynna ~ xo~anaxT-
Ha. 3Aecb MO}:CHO BOCnonb30BaTbCA o6~e~ TeopNe~f cc~epHUecxKx c~yHx-
uH~ (CC~,epYiueCKI~IX rapMOHHx) Ha KOMnaKTH~1X 0~[HOp.OAHbtx npoCTpaHCT-
Bax, pasBHTO~I 3.I{apTaxoM ~30~ H I'.Be~neM ~3I~ (cM.Tax~e ~IO, ).
PBCCMOTpYiM noJIHyIO CYICTe2Jly yHYITapHbix HenpHBO,~rMb~x HeaxBHBaneaT-
H~x npeAcTaBneHHi~ (2.6) rpynnm.~ H B~6epeM B~npocTpaxcTBaX
3TLIX npe,uCTHBIIeHLII~ 6a3HC TaK,uT06bt'OHY! pacnanxcb xa HenpxBOAx-
a~me npeAcTaanexHA no,~rpyrinm ~ . knA. Toro,uTOd~ MaTpxus~c~ ane-
~~~
a~eHT ~'~~. (~) 6~tn n0CT0AHH~c Ha Bcex neBnx Knaccax cMe~cHOCT11!
n0 ~ AOn}RHO B~InonHATbCA paB@HCTBO
(3.3) T,?~a~(k)=Sm~ ,~ m=1,2,..,d,, ~- kE~:
~TC~Aa BYiAHO, uT0 nOCTOAHH~Ie Ha nea~x Knaccax CMe~HOCTLi MaTpuu-
HI~e 3neMeHTbI Td~~~~) sanonHAIDT CTOn6u~t MBTpYIL[H T~"~(~ ~ , OTBe~18ID-~+
wHe eAKHPIUHHM npeACTBBneH-IAM ~ . IIyCTb,HanpHMep~ =;npeACTaB1Ie- 1'
4
HHe T ~'~~ rpynn~ ~ paBHO Za paa coAep~HT e~YIHI2~IHOe npeAcTaBnexxe `
1
. IIpeAnono~HM, uTO B HameM 6a 3HCe el, eZ, ... , ed~ 3Tx eAH-
HyruHm Via) peAcTaBneHHA oTBeua>oT nepB~ Z~ 6a3LICHm~a BexTOpaM (Tax
uT0 ? I (k).e~ = e~ npH k E ~G H ~ =1, ..., Za ). B TaxOM cnyuae~ ,"1
c~yxxuHx oT x
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 w'.^ a
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
i=1, ..., d?~, ~=1,..., Za , a=2,z,...
6ya~yT Ha3bIBaTbCA CC~epYI~eCKTIIMx . HK HAb1Y! Ha ./~ , a c~yxKuHx
(3.5) ~~ ,~ z,) = T CT )l ~ ~ , i .1, ... , ~Za ; ~ : 1, ---, z.a , ' ~ ' 1, 2 , .. .
.d d
- 30Ha1IbHbIMLi C~_YI~IeCK1lI? c~yHKuYiAMYi. nerKO . BYIAeTb, uT0 30Ha]Ib-
H~e c~:epr~UecxHe C~yHKuYTY! npHHHnaacoT nOCTOAHHbie 3Ha~IeHYiA Ha BCBX
,I;ByCT0170HHYlX xJIaCCBX Crde~:HOCTY! ~a~ rpynTlbl ~ n0 ~ ;
HHa~e roeopA, AnA xi~x ~?~~~(x) = ~~'(k.~) , k ESC ~ CoBOxyn-
HOC Tb TO~IeK ~x , jC E ~ , eCT6CTB8HH0 H03B8Tb C ~e 03~ C ueHTpom.
B To~xe xo(= ~~ , npoxoAA~e~ uepe3 .TO~txy x ; Tax~ra o6paso~a,
C~yAKL(HYt (3.5) nOCT0AHH1~ Ha BCeX c4epax c r~eHTpou B xo . IIo3TOaay
30HaJIbHaA C~y~iKLjLlA ~`~,~x~ c,~aKTYi~IeCKH., 6yAeT 3aBHCe.Tb JIYIIDb OT
d~
coBOeKynxocTr~ ~xBapuaaTOB ynopAAo~exxo~ nap~a To~eK x x xo,co-
XpaHAYJII~YIXCA npx npuu~eHeHHH x o6oHM TO~IKaM n~6oro npeo6pa3oaaHYlA
E ~ (OT C1IO~ HOrO aCCTOAHHA OT x AO ~o ):
~T) ~ - ;' .
(3.6) ~ , ~x~ _ ~`(~-f(x, x.) - ~`~ (~z, ~x')~ ~ E ~J , ~ i =1,, ..., 7a , .~= ~L,-.
d ~ ~ ~ ~~ ..
Cornacxo o6u~e~ TeopHH cc~epH~ecxr~x c~yxxuu~ c~yxxuHx~'(3.4)
npeAcraBnAwT co6o~ nonxy~ opTOroxanbxy~ cHCTeMy B npocTpaxcTae
2
~ Cx~ c~yxxuxi: Ha X c xBaApa~TOM MoAynA~, HHTerpxpyeubua ~ no mepe
C~x , LIHBapHaHTH03~ OTHOCI7Te1IbH0 BC@X npeo6pa3oBaHx~ ~ E ~
TOJIbKO ~TLI (~yHRI~YlY[. (3.4) H BXOAAT B pa3no~exxe (~yHKI(YIY! ~~~~ ~
nOCT0AHH03~ Ha BCeX neBbuc KnaCCBX Cl~e}~HOCT1d y/.~ ~ ~ ~~p
aeaTpx~rH~ 3neMeHTa~ T~~"~~~~ ([!o] , rn.Y, ? 23).
IIpr~exeHile 3TOI~ TeopYII3 x OAHOpOAHb~d Cny4a~IHb[M acnAa~ ~ (x)
B cHny Teopeun I npHBOAuT x cne,~y~~eMy pe3ynbraTy:
Teoperaa_5. A~ T, 1'O,~T^06bI cnyya~rHOe n, one
~~) ~ ~ .:, ~ ~ ~~ ~~> > ~ ~ f fi(x) ~' ~ Cx) f ~ ~ ~ ~x~~
d x ~ x d
t sr
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
~ i~fi ema a .eruu ~~,. '?`?'- -----
? -I9-
Ha KOMnaKTHOb1 OAHOp0AH0~: n OCT aHCT ~ ,x = g/,~ 6N0 OAH~OA-
x~e Heo6xoAH1,~o H AOCTBTO~IHO, ~IT~ 061 CJIy~0~1H~te BeJI~~I_~ ~~L ~
? yA0B1IeTBOpAP.YI COOTHO?eHHAM ?
(3 8) E~ ~"~~ rte, = S S. (a~ .
~ ~! CR ~~ ~K f JQ
~Y! :3lT01:9 KOPPelIAuMOHHaA Hx Y!A .nJ~.x1~".xz
CT8B1IATbCA B B~~B
,_
nonA ~~~ 6 AeT n~-~
rAe ~'~ (x, ~ xz~ - Hx HH (3.6 06~aTHO, n~o__ 6aA Hx Ha .~
..93(x,,xt) B Aa (3.9 ,rAe ~ f e"~~~ - 3 2~9bZTOBO HeoTpHl(aTe1IbH0 on a e-~
ne.--~ ~da~T Hu~I Tam, PTO ~ ~s) cxo LITCA, 6 eT x p enAIIHOH-
Ho~1 ~ Hxul~! ~ HexoTOUOro oAH~oA_____~HOr_o n_ OVA Ha x
~op~yna (3.9) npeAcTaBnAeT co6o~ xecxonbxo ycoBepmexcTBOBaH-'
s
Hym 3anHCb peaynbTaTa, cc~opa~ynHpoBaxxoro eke 'B I94I r. boxxepo~
~II~ ~IacTHi~e cnyuaeM Teope~an 5 AanaeTCA Teopema 06yxoBa ~6~ 06-
oAHOpoAH~ax nonax xa ABya~epxo~ cc~epe S~, _,. ? yno~aHHaBmaACA Bo BBe~ceHr~
x xac~os~ige~ cTaTbe. B 6onee o61~e~ cny~ae oAHOpo~tHnx none Ha cede-
pe ~,,., n,-aeepHOro eBxnHAoaa npocTpaHCTBa Hama Teope~a osHa~aeT,
PTO xax~oe Taxoe none 6yAeT pasnaraTbcA B pAA no rHnepcc~epu~ec- '~
xxa~ rapreoHHxa~i y ~ (B 8 `~~ e= o i z o~m ~ art, ~ .... m, ~e
(Bnpa~ca~lAxaecA ~epea TpHroxo~ieTpvi~ecxue c~yxxuizH H c~yHxuv~H I'exeH-
6eyapa oT c~epH~ecxHx xoopAHHaT B, ; ..., An_2 , ~ ; c~a.~32~ , T.II) c
HexoppenHpoBaxH~H xoa~~u~HeHTaMx fie, m., . , mK ~~` ~n.L , Ay~c_
nepcHA ROTOpHx 3aBHCZiT nHob oT a 3oxanbHne cc~epv~~ecxHe cpyHkc~ti~
B 3TOM C7ly~ae 6yAyT AaBaT~ n~, nOALiHOMatdY I'ereH6ey3pa ("ynbTpBC(~e-
pH~ecxvnar~ n01IYIHOMabdY!"~ C y (~, OT ~ . ~oSA; no3TOMy (~Opmyne
(3.9) 3ReCb 06paLja;'TCA B yI3BeCTHy~ (~opbiy,~y 1ffeH6epra ~7~ AlIA nono-
~HTenbxo onpeAenexx ~:~ gyxx~H~ Ha ~ m.-i~-~epxo~ cc~epe:
'1~ ? ~ ~ /
C=o
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 }~.~~' N Il it v. ~? '1 r t~ fi M1~..7
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
.. - - ~V -
~:NTenbxo onpeAenexxo~ c~yxxuHN ,~,(,x1,xZ)Ha X , yuoBneTBOpA~r~e~
( 812- yrnoEOe paccTOaxxe ~ae~:Ry To~xa~aN x1 N xt c~epn sm_i )?
TO 06CTOATeJIbCTBO, ~ITO B 3TOM Cnyuae C1IOb:HOe paCCTOAHNe ei2 (KOTO-
p0e 3AeCb 3aAaeTCA O,~HYi~d ~IYICTiOM) CYIMMeTpYI~HO 3aFYICNT OT TO~ex ,zi
N x2 (T8K ~ITO ~f0~:H0 TOBOpMTb npOCTO 0 paCCTOAHYIYi ~ae~c~y AByMA TOU-
KaMyI) x TO PTO Z~ ~1 nPN nro60~ /1 YIDleeT o6~gee o6~ACxeHVCe, KOTOpOe
6yAeT ACHO y[3 AaJIbHel~mer0.
Ilepe~iAe~e Tenepb K Cny~am IIOKanbHO KOMnaKTH6uc OAHOpO~(HbIX
npocTpaxcTB X . Ha~Hea~ c saAa~N od onpeueneHNN o6~ero BNAa nono-
ABnAeTCA HenpNBO,ux~~, c~yxxuNA (2.24), r,ue ~o yAoHnezaopAeT (3.I~
B ~acTHO~ cny~ae,'xorAa yHNTapxoe npeAcTaBneHNe T($) rpynn~ ~ ._
~TO6~i BexTOp ~o yAoBneTBOpan ycnoBU~:
(3. I2) ~'(k)~, -~o npN acex kE~C
BC@X AByCTOpOHHIIlx KnaCCaX CMeRHOCTY! no~G xeo6xoAx~o H ,I(OCTaTO~HO .~
lleHHhIX c~yHKuYi~ Ha J , nOCTOAHHI~X Ha AByCTOpOHHYIX KJIaCCaX CMe~HO-
C TH n0 .1V ..
B CNTiy [I7] IIIOdaA n01I0~YITeAbHO Onpe,~eneHHBA c~yHKuLiA Ha
3aA8eTCA (~OpMynO~ (2.24). AJIA TOr0,~T06b1 OHa 6biJIa nOCT0AHH0l~ Ha
3oa~ Hama 3aAaua caoALiTCA x pa3b~cxaHNm BCeX TIOIIO~YITelIbHO onpeAe-
Ho onpeAeneHHy>;o c~yxxuN~o Ha X , y~oHneTaopAm~gy~ (3. I). Taxiu~ o6pa-
ycnoBNb (3.I). KaacAo~ Taxo~ c~yxxuNN ~ao~:xo cne~y~u~Nr~ o6paao~ oAHO-
3Ha~H0 COIIOCTaBNTb IfOJIO)It1llTe1IbH0 onpeAenexxy? c~yxxuN~o ,~(~~ Ha J :
. (3. II) ,~~~.~=.~ (xi, xa~ , ecnN ~ - ~G,~Z ~, ~CZ , ~ xl-~,~ , x2 =~Z-~ .
O~eBNAxo,~TO npN 3TObQ .~(~)-.93(k,~kz) ,npH n~6~c k:, kti E.7~,
T.e..~S(~) npNHi~aeT nocTOAxHne 3xa~eHNA Ha Bcex ABycTOpoHHNx .xnac- ..
cax ca~e~cHOCTN y no ~ 06paTxo, nonox~NTenbxo onpeAenexxo~ c~yxK- ;
uHN Ha ~ , nocTOaxxo~ Ha Bcex ABycTapoxxNx xnaccax crae~HOCTr~ no
~bt uo~:e~ npH no~aoRH pasexcTBa (3. II) conocTaBNTb nono~NTenb- .
6yAeT aoxanbxo~i cc~epN~ecxois ~yxxuNe~f, oTDe~a~u>,e~ aTOr~y npeAcTaH
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 .. _...
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
.rua 6T~u ~ .~,~ ~:m a
~..~.o ~E;E'v
- 2I -
neHVI~. Boo6vge, ~yHxuvlA . ~'(x) xa3xaaeTCA 3oHanbxoi~ cc~epH~ecxo~
c~yxxulze~t xa X , oTee~Ia~otue~ HenpHeo,I~rIMOI~y yHHTapxorRy npeAcTaBne-
HLIiO T~''~(~) , ecnH oxa npeAcTaaHUa B BHAe .,
(3. I3)~(~c) _ `~(x~,x)=(T ~'~~(~)~, 7), T~''(k)~= ~,T~'~"k)7:7npH acex ~kE~
I4 Ha3bIBaeTCa npocTO c~epl~~ecxo~i c~yxxuvle~, ecru
(3. I4) fi(x)=(T`'~'C~)~,7), T`~~(.~)~= ~ npvl Bcex kEJ~G.
O~eBHAHO,~TO 30Ha1IbHble cc~epvl~ecxxe e~yHxuHH (3. I3) .3aBHCAT nHmb
OT Cn0~CHOr0 paCCTOAHLIA OT x AO .zo , a c~yHKi(LiYI (3. I4) - OT TO~Ixvl
pasnaraeTCA B xo~TVlHyanbxy~o npAMy~ cyI-~y xenpHBO,~IrII~I~c He3xeHBa-
neHTH~x npe~CTaEJIeHYiI~. ECJIH ~o yA0H1IeTBOpA2T (3.I2), TO Tlpoex-
cy~,y oAxoxpaTxxx npeAcTaBneHH~, xa~uoe vIs xoTOp~Ix B cao~ o~epeAb~
IIpeAnono~I~u Tenepb, ~ITO rpynna ~ ~ - cenapa6enbaaA noxanb
Ho xounaxTxaA rpynna ?_Tta na I. B Taxo~ cny~Iae a cHny ~2I~ , ~22~
~~ Mo~el~ pa3nox~IZTb npeAcTaBneHHe T(~) ~ e~op~yn~ (2.24) a npA~ay?
LtYiA 3TOr0 B@KTOpa B npOCTpaHCTBO IIto6oro HenpxBOAl~oro npeAcTasne-
yAoaneTBOpA~i~aA (3. I), 3Aecb 6y,~eT npeAcTaeH~aa c~opMyno~
HHA T `~-)(Q `
4 J , HxoAaiuero a cocTas T(~) , 6yueT LlHaapHaHTHI~d BexTO-
pota cooTEeTCTaybiuero (npHeoAl~oro) npe,~cTaeneHHA T~''~(k) rpynnx
. Paccyy~AaA Aanee Tax ~e xax npvl BxBOAe ~opneyn~ (2.26) ~~t
Ha~;Ael~, uTO npoH3BOnbxaA nono~HTenbxo onpeAenexxaA c~yxxuHA Ha x,
~~). --II
paHCTBO H~ , ~~`u~~ - 3pMYlTOEa HeoTpLIL(aTellbHO OnpeAelI8HH8A
"onepaTOpHBA I~epa" Ha ~ CO 3I?a~IeHYIA.~Li u3 KO1Ibua OnepaTOpoB B
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
rAe ~ - no,I~axo~:ecTao Tex a ~ J , unA xoTOpxx B npocTpaacTee
H`'~~ b~~eeTCA XOTA 6bt O,d,LIH BeKTOp, YIHBBpYiaHTHI~1~ OTHOCYiTe1IbH0 BCeX
npeo6pa3oaaHVl~ T~"'(k), kE~ 9'x~, - onepaTOp npoexTHpoBaHHA H H`'~~
Ha M8KC1iManbHOe YIHHapxaHTHOe OTHOCHTe7IbH0 Bcex T~'~~(k)noAnpocT-
 Approved For Rel@ega~se 2p0~0}9~/(07/09 :CIA-RD~~?P8u0Tg00246A011700340001-4
- 22 -
npoCTpaHCTBE .TLS, , a ~s H Cjz - npol'~3FOJIbHF1e 3IIe~eHTbi Ilia
xnaccoB c~e~::HOCTH n0 ~ ~ 3aAaroAYIX TO~IKH ,~1 H xz x3 X ?
OnepaTOp g'x~'TU~(~)g'z' B c~opMyne (3. I5) ecTecTBeHHO.pacc~aaTpB-
BaeTCa Kax onepaTOp B noAnpocTpaxcTBe ~~~ ; xHTerpan no y,~__
3,~eCb CJIe,I~yeT noHYiD~aTb K8K CyrnMy YIHTerpanoB, paCilpoCTpaHeHHbUC n0
nOAhiHO~:e CTBa~e ~w ~ ~~ ~ rL = ~ ~ 1, 2, ... TaISTrIM, ~ITO npx .~ E yn noA-
(1-f
ITpoCTpaHCTBO H~ ABJIAeTCA ?lrbdepHblM.
TBKYIbi 06pa30M, TILMeeT McCTO CJIeAyIDII!~ ~eo~e~a nepenxcaTb c~op-
Myny (3.I5) B BxAe
~~-)
~z ~~`)
HenocpeAcTBexxo odo6u~awuer~ (3.9). 3~ecb ~epes ~~~~ (x, ~ ~~} o6oaaa-
~exo noJIHOe ce~ei~cTBO nHHe~iHO He3aBZSCt~bix 3oxanbxbix cc~epH~ecKAx
c~yaxuH~, oTBe~a>:o>uHx HenpiRBOA~9oMy npeAcTaBnegHio Tt"'(~) ~ ; 3To
CeIde11CTB0 6yAeT KOHe~HHM ( YLL-~~neHHbua) nplz ~ E y,L, 7L = 1, 2, - ? ? ~~
H 6eC$oHBHi~d npx a F yo, . - ~7
CygeCTByeT eiue O,~HH Ba~HF3~i KnaCC OAHOp0AHb13C ^poCTpaHCTB,A1ia ';
xoTOpnx ~ao~:HO Bi~inHCaTb o6>r~yio c~op~iyny (iz npHTOM Aare 6onee npocTygj
~IeN (3. I5)) Ana npoH~ onbHiaix nonoxcxTenbxo onpeAenexx~x c~yllslix~
,~
~,~ ~~ ;~pn~r n a RA~~ ~~nn ~n
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4 -
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
m m~mr tam m m~~a~~m. ~~,. ~,?~.,
- 23 -
HLI~ YIMeeTCA NHBOII~TLiBHbi~1 aLT02:fOpt~N3M L~-'C~' ~T.e.YI30MOpC~HOe OTO-
BO .l1 "' ~/~ Ha3bIBaeTCA cNMMeTpN~ecxl~aa, ecru B rpynne ~1 er0 ABHZ~e-
~ecxvie oAxopoAHr~e npoCTpaHGTBa 3.I~apTBHa. O,~HOpOAHOe npoCTpaHCT-
.~(xs,xz) , yAoBneTBOpAiflti~Nx (3. I). 3TMM xnaccoM PBnA~TCA cvIMMeTpH-'
dpaxceHVle y Ha J , npH xoTOpot~i (~~)' _~ ), Bi~iAenAtaluN~ cTauvlo-
xorAa ~ E ~ HeTpyAxo BizAeTb, ~ITO 3TOMy ycrioBNio 6y,i>vyT yAoBneTBo
HapHyio noArpynny T. e. raxo
ITO ~ - ~ TorAa N Tonbxo TorAa,
~ ~_ I
~
pATb, Hanpl~~ep, 7Ii061~e OAHOp0~H1~Ie npoCTpaHCTBa noCT0AHH0L7 KpYiBH3-
Hbl ~Cbi.,HanpI~Mep, ~33~ ,rAe MO}.?HO Ha~ITY! TaK~Le pAA Apyrxx npvl~ae-
xHHry ~I3~ , ? 3I) 6r~ina ,uoxaaaxa cne,~yiflutaA Ba~HaA TeopeMa,cylge-
HopOAHbiX CI~iMMBTpYI~IeCxYiX npoCTpaHCTB reJIb(~iaH~O1~I [34~ ~CM.TaK~e
poB O~HOpOAHbI3C CI~IMMeTpYI~IeCKYiX npoCTpaHCTB). ,~'1IA npoI13B01IbHb1X OA
Teope~a_ 6' . B c____ny~a_e cNMMeTpN~ecxoro o~HOpoAxoro >~o~axcT-
Ba x = ~/3G xa~c oM HeripNBOANMO~iy yxx~a HOI~4 npe~aB__ nexNro 'j'~"~)
CTB8HH0 ,AOnoJIHAIDIijaA Teope~eyy 6:
r~ynni~ ~ oTB~ He 6~ ~~ o Ha aoxa~~ibxaA c~epN~ecxaA yxxuvlA
l~+) 1 la)
(xs, xzl , T.6. TIOAnUOCTpaHCTBO H~ ~ 1II060~t ~1 E ~ 3 e b
H6 6~ a ~I~ OAHOMe H0. n, 6aA YIHBapYiaHTHaA 0_ THO TeJIbH9 BYI~{ HYI~!
nOlIO~YITeJIbHO OnpeAelIeHHBA HK YIA ~(.~~ ~ xzl xa Ta^KCM npO.CTDaACT-
se X Mo, ~_eT .bib npeAcTa_ exa "oc,~ M~Ofd
rAe ~(d.~) - HeoTpNuaTe~ibxaA Mega Ha ~J~ (coBnaAaioi~e~e B 3TO~e Cny-
^e C ~~, ) Ta~A,~I^0 LIHT~~1I B n~?~ ~Ia~ CTLi ~3.I7 CXO YiTCA.
B cny~Iae cvIMMeTpN~ecxoro. npocTpaxcTBa X c~yHxuHH ~~~I(xl~,x,,~~
6yAyT CYIMMeTpvi~H0 3aBViCeTb~ OT .x~ LI rz ,TaK xaK 3AeCb BcerAa cy-
IAeCTByeT ABI2:~eHI2e, M6HA~JIi~ee nopAAOx 3TYlX AByX TO~Iex, ?llHdye ~0~0~..st~
OHLI 6y~jyT 3aBNCeTb JIN1Db OT CJIO~fiHOr0 pa CCTOAAYIA M_Q1{,1131 TO.-~x-~yi xs
H x~, (cl~. ~33~ ) .
~e~A~~ ~~~ ~~~_Y
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
~. r~~i ~~ ~.'~2~~--~~~ ~~~LiI
. IIepe~Aer~ Tenepb K "CneKTpaJIbHOMy pa 3JI0:teH3~iI0" Cardor0 OAHO-
poAHOro cny~a~HOro none ~ (x) . TlpeAnono:ry.vl~, PTO xoppenauxoHHaa~
(}7yHKuLiA ~~x~s~ xL~, 3TOr0 n01IA l~o~{eT 6~1Tb npeACTaBJIGHa B BHA@
(3. I5) (uaczH~a cny~ae~ aTO~ c~op~ayn~I ABnaeTCa, H (3.I8)~. byAe~
paccymAaTb axanorvl~xo Aoxa3aTenbcTBy Teope~x?".. Pace~oTpvlae rHnb-
6epsoBO npocTpaxcTBO L2(3,~) onepaTOpH~x c~yHxux~ ?.~~~, ~ E ~z~
co axa~eHHal~x yIS xonbua orpaxH~exH~x onepaTOpoB, Ae~icTBymntux xs
Hx ~ B H ~'~~ , H Hopmo~
(3: I9) II u`~'~ 2 =1T.~ ~uca) ~~ (d.~) U~'~~~~:
JI~6o~ orpaHH~IexH~~onepaTOp, V B N~'~~ ~x Mor~ea~, ecnH yroAxo,
paccMaTpvlDaTb xax "onepasop ~~ Hx' B H,__, ",, orpaxH~HB ero 06-
nacTb onpeAenexHa noAnpocTpaxcTBOr~ Hx~ ; Bo xs6e~caHxe nyTa~xux
~~
rn
~
3TOT "cyx~exHxl~" onepaTOp ~~ 6yRe~a. o6o3Ha~aTb ~epe3 V JjC ~ .
B Taxol~ cny~ae cooTBeTCTBiae ~ . ~ ~, ~ - .
(3.20) T. (a)~~~~~,~~'-' ~ ~x~ "Pu x.= ~~ ~ ~ ~ .
~
. ~~
.
6yAeT i~3o~eTpH~Iecxl~ra oTOdpa~eHHe1~ ~aHOa~cecTBa ~~(x),.xEX}npocTpax-
CTBa '111. B ~Z(~7GJ , KOTOpoe ~0}~HO 17p0A0A}~i~Tb AO YI30MeTpH~eCKOrO
OTOGpB~eHHA ~Z(3x~ B ~ ECnH TeTlepb nape BBKTOpoB ~IEH~"; ~,E Nx~
H YI3BdepvinaOluly NHO~-eCTBy ~E ~~ Mbt GOnOCTOBI~IM ll~"~11; ~1r7~'~ E +2(3x)
c nor~owb~ ?opa~ynx (2.30), To npvl oTOdpa~ceHxK (3.20)
(3.2I) u~a~~n ; pl' p~l'_'.~~n j pig p~1 _ (~ ~~~ Ts,T~~ ~
rAe ~ ~11~ - cny~a~iH~i nvlHel~Hbl~I onepaTOp Hs H B Hx , saBHCa-
iqH~`~ oT 11E y7c, ranee Tax sae xax npvl BxBOAe c~op~yn~t (2.32). Aoxa-
s~IBaeTCa,uTo
(s.2z) ~~x)=JTz(~(da)T~a~~~)~'~'~~), x={g'7C}~~
~x pp
x ~ITO npH n~61~x ~, , dl E H ~~~ ~ Tz ~ ~~, E :H~a)
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
R~ ~
~~ ~ ~~~? m ~~~ ~
(3.23) E~~~~1)~1,~~)(~(~L~~1i~~) _(~,,~1~(.~,-~~l~i nnZ)~,~, ?~.)~ _
OTCwAa nerxo nony~aeTea cne,~y~iuaA . - -
Teopeld~ 7. nA TOrO, ~IT^ 6~ CA. ~1HOe n~ ~(x) Ha OAHOpOAHOB~
i
n OCT aHCTBe X= ~/~G , rp~~ ~ ABI~'~eHYiI~ KOT~ Op' Oro AB1I~ AeTCA
Cep 8Aa6enbH01~ nOK---mot K01`! axTxo ~ r nnoii TM~nc `I~, 6^ o OAHOpo,I~Hb1Ed,
He, 06x0 HrdO YI ~OCTai0~H0, ~IT`~I~ e~'O I,~O__~_~".H_0 6bIJI0 llpe CTBBHTb B B~YiV
3.2 , ~ ~(~~ - cny~a~HI:~I~ nHx~ on^paT~po ~ H~"~ ~ .N[~-I
aA~YITYIBHO 3aBYiCAIi(YIYI .OT MHO~:CeCTBa n c ~~ yI y,uOBneTBOpAwrl~
n10- ~-~~~~1 E H~~~, ~z~~zEH~~ ycn~ 3.23 3A~ecb .~x(11~- 3~
Tti Heo,~nH TenbHi~ one___ paw B H~ ~ ~ yAo` BneTBOpAIOUIYI~f (3. I6 , ge-~
KOTO ial1~ KOppe1IAII110HHHA C~ HK HR ,~3(,xl, ,xy, TI, OJIA, ~(,~) Br~pa~a eTCA
nix nor~a01'n opM n~i 3 I5 - ~ - -
AHanorH~x~Ie e3 nbTaT~a cn aBe nHBi~t H nA o Ho o,uHi~x cn ~a~l-
p- yv--- --p- ~_ ~ ~ ~.._-__.., .-.Y.__
~ nom Ha npoH3~ bHOb~ Cl~IM! bi~I~ eCK01+d o'~xopo,I(HOM np~ aACTBe
X , C ~ TO~~ a 3HY!-eY , ~ITO B 3TOM Cn~ B npoCTpaHC TBO H~~
o~HO~~epxo ~ nlo~ ,~ E ~y~= ~yl ~, cne osaTenbxo, ,~(~) -- cn^y~a~~
H~~I nYlHe~IH ~ ~ xxuuoxan B H j"~
~, ,_____~ ~ , yAoBneTBOp~iolgH~i y noB1R~
(3.24) E~(n,) ~1 ~~~z~Q1 - ~(ns n~t~ ~~_ , s~,
. d ~
rue ~(~ ~ - xeoT~pxl~aTe~ HaA ~-ae~a B ~y} , ~I~r pHpymu~aa B c~?~ne
3. I8 .
B "xoopkHxaTHO3~ 3anxcH" gopluyni~ (3.22) H (3.23) 6y~yT, pa3y:
i~ee.TCa, Becb~aa noxo~H Ha (3.?) x (3.8). B ~acTHOCTx, B cnygae
clzl~meTpvl~ecxoro .npocTpaxcTBa X 3TH ~opuyni~ o6paII(a~TCA B
(3.25) ~(x~ _ ~~ ~,~'~'(x)~~,(dJ-), .
-L
s ~'-~' 1
(3.26) E~~ (nil ~m l~z/ = SnnL J(~i(lnz~~ -
(x)
rAe ~n (x) ~ n,=i,2,..} ? nonHaa cr~cTer~a "c~:epx~ecxHx c~yaxux~", oT-
(a)
Be~a~>:~Nx 30HanbHO~ c~~,pH~~,~~,~,I~,~~c~Ku~ ~ ~xs.JCZ~ . TeopeMy 7
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
'~ ~~
(BMecTe c Teoper~ardH 6 is 6~') MO~:HO p?CCMaTpYIBBTb xax o6o61~eHVie
npHBeAexHi~x Bbune~ Teoperd I, ~ 3, 4 x 5.
II rarde ~a. I-i3BecTHO,~TO a n-rdepxoM eBxnH,uoBOM npocTpaacTBe
R,n (c o6~~HO~I rpynno~3 ABI~~:exvl~ ~ ) xr.~ieeTCA oRxonapaMeTpvl~ec-
~"~ sa auc A>luHx
xoe cel~e~,cTFO 30HaJlbHbIX c~epK~ecxrsx c~yxxuH~ ~' ('~~ (
IIYIIDb OT paCCTOAHYlA Z rde::CAy TO~KaA4Y! xs H .x2 ~:
(3.27) ~C~, `i -~~ ~~~~ .T ~, Z )? D L ~ L ~ ~
C
(cri.,Hanpl~ep, ~35~ , ~28~ ).. Tax12r~ o6pa3ord, . c~opuyna Illexdepra (L4)
AansleTCa ~acTxbird~ cny~aerd o61ue~ ~opmynl~ (3.I8).
. 06i~iRe c~epu~ecxvte e~yxxuHU Ha eBxnHAoBOr~ nnocxocTVl R.z Br~rlrzca-
H~ B paGoTe Kpe~iHa 351 (cM.Tax~e BranexxHx: ~24~.).. TIoAcTaBHB 3TYI
c~opr~yn~ B (3.25) - (3.26) ral~ nony~~Ird cne,~y~~tee o6uee npeAcTaane-
HYle O,II~IOpOAHbiX H YI30TpOnHbix cny~Ia~Hblx none~i Ha nJIOCKOCTYl:
(3.23) ~(z,~P) _~ e eJn (z.~)~w(da),
n_-~ f ...
rAe (z,`~) - n01IApHI~Ie KOOpA1~iH8T~I Ha nnOCKOCTYt B ,~n,(da~y~OBJIeTBOpA-
eT (3.26). AHanOrx~IHO 3TOMy AJIA O,I~HOpOAIihIX H yI30TponHblX n01ie3~ B
11rMepHOM eBKnHAOBOrd npOCTpaHCTBe R,,,,11lCXO,t~A Yi3 pe3yAbT8TOB 3aMeT-
KYi f 281 (CM. TBK~Ce 321 , TOM II) Mo~ceT b~iTb nony~erlo npeAcTaBnexxe:
(3.29) ~ C'Z, 81,..., Bn-2,~~'
~e,~~~-)
EA Z (~-)~
Q,mt,..y mw?S,t nlh.1 e,Ylt,,..., T11x.,,imw.y ~~ 1 `~ ~ i! ~ ~
rAe Z , Bs, .. ~8n-z, ~ - . cc~epvl~ecxHe xoopAizHaTx B fLri , cyMMHposa-
xvle B npaBOl~ ~acTiz pacnpocTpaxexo no BceM ~=0, 1,2,...,o~-nn,,=m?.,~ _..
... -`-ns-`~ ~I3 AHyM 3Haxala rnn-Z ~ A~e,ms,.:.,m,,.,,sm?.,, - xopa~t~poaa~-
Hble KOHCTBHTbI, i1pOCT0 B1;Ipa~aIDIl(HeCA ~IePe3 r -f~yHKI.~H~,~~m,~~~n~,,.~~:n~~.~,
- COOT92TCTBy1BIl(Yle IIOBepXHOCTHbie I'BpMOHflKH, a
( ~- C~IeTHOe Cerdei~CTBO B3anirdHO H@KOppenHpoEaHHF~IX cnyga~
~Z,M,~ . mr.~tmn.y ~ .
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
~ru~~ ~~~ ~~~~. ~~~ ~~~L1
- z7 - .
Hxx reep H8 npA2,d0~ [Di ?O~ C O,II;LIHaKOB~Ib! b1aTel~saTYl~IeCKI4Jd o~H~axHe~e
xBaApaTa Mo,~yna.
B cny~ae n-~aepaoro npocTpaHCTBa JIo6~~eacxoro ~?, 3oxanbH~e
c~epH~ecK~ze e~yHxuKH x~ae~T BHA ~ .
a .
{c ~.z)
J _
rfie ~~ - cneuxanbHOe pemexHe AHC~:~epexur~anbxoro ypaBHeH~ca
JIe~~aHApa (cb~.Kpe~iH ~35~ , IixnexKxH ~29~ ;Ana Tpex~epxoro npocT-
paxcTBa Ao6a~eBCxoro, rRe ~~T~('~) = S"~ T-' r ~ , cooTBeTCTBym-
a-ss~.z
r~H~i pe3ynbTaT eke paxbme 6~tn nony~eH I'enbc~axAoas H Ha~r~apxo~c ~36~ -~
~~8~ )..OTC>aAa AnB xoppenauHOxxo~i c~yxxur~v~ x3oTponHOro nonA ~,?,
nony~aeTCa cneAytoutaA o6u~aa c~opMyna (BnepB~e yxasaHHaa Kpe~aHO~
o
B pa6oTe ~35] B~nHCaH~t Tare ace He3oxanbx~e cc~epHUecxue
~yHxuHH npocTpaxcTBa ~Z ; noAcTaanaR izx B (3.25) ~~t nony~x~e cne-
,~y~n~x~ o6~H~ BHA oRHOpoAHOro cnyua~ixoro none Ha ~nnocxocTH JIo6a-
~eacxoro:
(3.32) ~(z,~) ~ one )~'1~ Y-a ~C~,z)~~(da~ ~ . ~n: (-i) ~
r,~e ~n ~da} yAoBneTBOpawT (3.26). Ana n-~epxoro npocTpaHCTaa
~,,, BCe 30HanbHbie C~yHKuYlH yKa3aHbI B 3aMeTKe f 2~, ; OTC1C~a Ana npo
H3BOnbxoro HaoTponxoro nonA B ~ri nony~aeTCA npeAcTaBneHHe Tuna
(3.2 _~,~ B KOTOpoM TOnbKO (~yHK1.~YtIO ~e' ~z~, cne~yeT 3aMeHY[Tb Ha
~
Z~ 3HaY (2 a) L
~~
~ ~ (H yI3MeHYIT~
HCTBHT
PaA ApyrHx npx~eepoe nonH~x cHCTea~ 3oxanbH~x c~epu~ecxHx
~yHKuu~ Ha ~aCTH~rx OAHOpOAHbIX nr)oCTpaHCTBaX yKa3aH B pa60T8X ~26~
~37] , j38~ ; B ~33~ Yi3y~eH~ TaK~:e HeKOTOpble 061gYie CBO:fCTBa TaRHX
Cr~~ ~~'~'~.~' ~ ~ ~Q~~^ r~~~s ~o
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
art .t,ti R'?s_~p-
~yHxuxH, cy~ecTBeHHO o6ner~aiouivle rax Haxou~AeHVIe. Bonpoc o Haxos-
AeHI~H np0YI3BOnbHI:IX (He30Ha7IbHbIX) C~GepYI~eCKIf;X (~yHKuYi:3 ABJIAeTCA 60-
nee CnOz~HbiM ; OAH8K0, AnA KOHSpeTHb?X npOCTpaHCTB 11 ~ ~/~ OH TaIt-
~e B pAAe Cny~IaeB MOB:".eT 61~ITb pemeH BnOnHe 3(;~eKTHBHO.
4. ~~IHOro~sepHi~e oAHOpoAHi~e nonA~. .
~a1IbHeifL'll7te o6oGIgeHYiA.
IZoHAT>aaTpH-
(4.s) M(ss) = u(~)M(~ 9s) = u(~,)M ~ ~ ~. ~~ .
(~~.~)~( 3> L =u(a)~C_i41i~'~~+lu#~~)=u(Qy)~(?'(G^'ai~~*`~L~~ ~ _ .
~ ~~ ~
rAe M`~' - HeKOTOpbI~I nOCTOAHHI~'l! BeKTOp, a ?~~?~~~, - MBTpYlua, 39-
BYICAIuaA OT OAHOrO OpryMeHTa. AHanOrLI~IHO OnpeAenA2TCA K none ~ ~~,
OAHOp0AH0e OTHOCMTenbHO npaBblx c,uB>!iroB: B 3TOBd cny~ae HaAO TOnb-
KO nOnOhl~iTb: Vit ~((~~, = U.(~~ ~~~i 4/
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
v ~8? ~a a 4~e_~~2y~~~r E~~~~,1
s
CneRya Ko:IrdoropoPy (crd.PoaaxoB ~39~ ) I~dxoromepxoe none ~(~~
rdo:~xo Tax:r.e laHTepnpeTl~poBaTb xax 3aAaHHOe Ha ~ none ni~Hel~Hl~ix
onepaTOpoB ~~ (a) yf3 H6KOTOpOrO )II~IHeMHOrO npOCTpaHCTBa ~ B npocT-'
paxcTBO cny~ia~iHi"Ix Benvl~ixH ~ - _.
B 3TOZd cny~iae none ~~ (a) dyAeT Ha3biBaTbCA OAHOpOAHblyd OTHOCYITenb-
HO IIeBbIX CABHrOB, eCnvi CytueCTByeT npeAGTBBneH~Ie {U+(~)} rpynn>~a
B npOCTpaHCTBe ~ TBKOe, ~ITO AJIA BC@X ~, 41, 9zE
(~.s) E~ (a)=E~ 9~(u'(~~a)~ E~~~(al)~~~(a~)=E~~9~(u+(~)a~)~~~,~(u'(~)az) ~~~
~1 $
(npeAcTaBneHVIe 1,~.*(~~ cBa3axo c U.(~) caoTHOmeHVler~: l~*(c~) _
TaxaA "6ecxoopAvcHaTxaA" yiHTepnpeTaLtYiA ~HOroMepHblx none~t oco6exxo
yAo6Ha npiz Hsy~eHr~vl none 6ecxoxe~HO~aepHt~x:
eTCA alga TOJIbKO BbIACHYlTb 061uHN BHA MBTpYil(I~I .(air ~~,, (YIJIYi, 4T0
TO ~e cardoe, MaTpHui~ ~~~~(~~ ). C 3T0~ L(enbi0 2d~i AOnOJIHYiTelIbHO
npe,unono~Hra, ~ITO rpynna ~ - cenapadenbxa~ noxalIbHO xordnaxTHaA
rpytTna Tvlna I (B ~IaCTHOCTYi, OHa blO~eT 613Tb np0Yi3BOJIbH03~ KO~anaKT-
06luu~I BHA BexTOpa cpeAxlrx sxa~ieHVi~ oAxopoAHOro none BenH~HH
(o) ..
HenocpeAcTBexxo onpeAenseTCa c~oprayno~i (~.3): BexTOp ~ B
3TO~f c~Oprdyne M07KeT 6hITb JI~6bI2~d nOCTOAHHbird BexTOpONi LI3 ,~ ~(,'Ta-
Hod rpynnoifi). Ranee nyCTb Cl. - npOM3B01IbHbILi nOCTOAIiHI~i~I BBKTOp
Ins ,~{, LI Q,(~) = Ll.+(~.) a . TorAa B cHny (4.6) . cny~a~HOe none
6yAeT O,If,HOpO~HhIM OTHOCYITeJIbHO neBi,x cABVIroB OAHOrdBpH~ird nonera,
nxxe~xo aaBrlcai~Irll,~ oT napaMeTpoB C~.~, . IIpHaaeHHB K 3TOa~y nonic
Teoperdy 4, nony~xrd
y
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Rell~e6aFste 621009/07/09 :CIA-RDP80T00246A011700340001-4
- 30 -
rRe
~ ,~,
a ,~ ~~~ = ~s (n), ..., ~~,(~)~ BexzopxaA "cny~a~fxaa onepaTOpxaa Me-'
pa" Ha ~ ~ Taxaa, PTO _ -
_ ,
(4.I~)E(~~.(n,)~i~~z)(~w(n~,)~,,~a~ C~~,~l/l3mn.(ni n~y)~~,~L~) -
(4.IZj E~~,(~t) f 1, ~Z)C~&~n~)~1~~~) =(~1~~1)C~Cni~".i)la)~z)~ C~~ ~L~)?
3Aecb ~~n)=~J;,w,n,n,tl1)II - 3T0 aAAvcTHBHD 3aBYlCAII(YIIr! OT~ 3pMY!-?
TOBCKHI7 HeoTpYtuaTellbHbll~ onepaTOp c Roxe~HbIDe .cneAora a "xpoxexepoH-
,,
cxo~ npoH3BeueHr~v~" Ax H~'`~ npocTpaxcTB Q H Nr~-~ (T. e. B npRrao~
cyu~e xaoa~op~xb~x H ~~~ npoc Tpaxc Ta e, H ~"~, e,, H ~'~ ; ... , rAe ~, .Cs ~ ,
eL, .. ? } - 6a sHC B l~). ~op~,iyny (4.8) uo~cHO Taxxe nepenHCaTb e
(4.I3j ~C~~=1 uC~)TZ(T~~~C~~~Cd~))
Hnvt ~e
1
(4.I4) ~Tn~~~-L+umrti.~0~~~.T`'`'(~)~d~,w(da) ? -
y i~ a
PaaeHCTBO (4.3) npx 3TO~s 3RBYIHaneHTHO ycnoeH~
0 , ecnu ao c ~,
rAe {T~a.~(~)}- -e~HHYI~IHOe npeAcTaBneH:ie rpynn~ ~ ; Tax uTo
'r'~'`?~(g)= I. ~JiA KoppellSiLjYI0HH0~ 2yfaTpHl(~1 .PJ(~,~ ~L, Yi3 (4.8) x
(4.I2) BxTexaeT c00TH0IDeH1de - a .
(4. I6) .~J(~1,r~,,)=U~~,~ JTZITC~,~~z91)3Cda~J U~~~z~
y
= u(~~) ?u(gi 9,) Tz jTv-)~~~ ~,)3(da)}?U"(~~~
r=ran ~~~~%~ ~ ~ ~~C~~' rn?C~~~ aq
? Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
i~nYl HHa~e
(`~.I7) .~Mn(gi,~z) ~ ums(~!) unt(~L) ~Td~'~~L ~1~ d ~,st(~a)~
st -a
. ~a
~IT06bi 3T0 COOTHOWeHYie 6i~no COBPdeCTYIMO C (4.~), ,~Ona~sO
~nA TOI'O
,
TonbKO BbInOnHATbCA HepaBBHCTBO:
(4.19) ~Mw (a o) >: MM' M`~,' ~ ; .
OTCi0A0 nerxo BbIBO].1;PlTCA
Teopelaa_8_ ,~]IA T~,~IT~ I~xorordepxoe cnv~ai~HOe no;;e
z nc
Ha r-~e cenapa6enbxoli nOxanbHO . KOIv1naKTHO]riY ~ T_ Yana I 6~ o~HO-
0 Hblyd OTHOCMTenbHO n@_~biX CAB~r_OB n0~1d BenH~IYtA ~ ,n~6D83y1C-
Y[1~X~A n0 npeACTBBneHYItO ~u ~~ )} , He06X0Abgd0 YI AOCTaT0~H0,~IT~06_~_
Ho 6~Ino npeAcTaBi~I~o B~ (4. I3 , r e ,~ ~1~~ - HexTOpHaA "~r-
~aNH~A One paTOpHaA ltiie~a" xa / , yAOPneTBDUAIOII;aA 4 I2 , a ~(~l~ 3
- "One aT0 HaA Me a" Ha ~ , 3Ha~eHYIA KOT~O~ AB:IA~A 3 M~TOB-
P p _-.P~ ,,.. _-~.~... .
CK~Y! HeOTpYlitc Tenb bIrdY! OnepaTOpab4Yi C KOH_B~I~ CA_ eAOM B "K~ O~eKB-
OBP-.-~It-?~ npOYl3BeAeHYtYj" Ax H~~~ nAOCTpaHCTB A H H(am) CP~-
H?_e .3Ha~ xr~ee M = Eg(g) n^onA ~Cl~~ ~ 3,TOU BeTCA 0 M JIOd 4 3 ,
M~?~ onneAen_____AeTC_A_ ti (4. I5 , a xoppenAuHO_ ~IaT Y! a ,~(~~~z~
- ordnol~ 4.I6 ~ ~ ~ .
06~HO, nro6aA uaT H a Bea 4.I6 ABn~ xoppenAunoxxo~
I~daT iz He, oTO~oro ~dHOro~deuxoro oAxopogxoro cny~ oro_ nom;
C e Hee 3xa,.,`exHe aTOro n~ a~o~e nPHH~aTb n__ 6oe_ 3Ha~ BH a
(4.3 ~ rgg M~~~ - nOCTOAHHbt~I BeKTO y~I0B1IeTBOp 4. I9 .
($)Irixorol~sepHble O~HOpOAHble n~ , a OJ~,HOp_0~ IIpOCT__paHCTBax. y
rixorol,~epxoe none ~ (.x~ ={~1(ac), ~z(x),...~ Ha ~ _ ~/~ Ha3blaaeTCA
oAxopoAH~, ecnYl ero nepBi~e YI BTOpI~e ~ao;~seHTbl He rdeHA~TCa npH npYt-
a~exeHVlvt x Be~:is~I-xard ~~x~ Qpeo6pa3oBaHi~A ~~x~--il(~~~~j~~i~G~~
~ ~ ~ ~ ~ ~~ a~ } ~ ?5~~ ~t~g~ ~
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
~~~ ~~"~~~~~~~ ~~~ ~ic6oM k E~
(~.29)t~~(~k,)=~~~(~~~U~~(~~)_.~E~~(~)u~e(~)ue~~k)=~7Q(9~U~~~~)~
(4.30)~~,~ ~~~,~ T~,~~ (~)Tn(k.) =~~~~n~-a,LT-,U'`dl.uei (k) ?
Jl m,n, r ~- ~ L
OdpaTHO, Hs (4.30) cne~yeT, PTO ~~(~~)_~~(~~ ;TaxYr~ o6pa3om,yc-
1IOBLIe (4.30) AB3IAeTCA xeo6xoAv~Mbc~a H AOCTBTO~iHblal A1IA TorO,~IT06~i
H~u~ noneM Ha X= ~/~G .
none ~~~~B (4.26) - (4.27) 6~no Ha caMOM gene oAHOpoAH~M MHOror~ep
B CLiJIy COOTHOIDeHYII~ OpTOroHalIbHOCTYi (2.8) Yt3 (4.30) BbITBKaeT,
uTO npH n~6~x ~l,m,..u rt, ,~na Bcex k E ~ .
B~6eper~ Tenepb H B npocTpaxcTBe .~ , B xoTOpo~s Ae~cTByeT npeAcTa-
BneHHe 1.~?(~~, H a npocTpaxcTaax trpeAcTaBneHH3i T~~~(~~ ~, ,~1=1~z~...~
da3YlCbt TaK1lM o6pa3oM,~TO6~ 3TH npe~cTaBneHi~A pacnanHCb Ha~ HenpH-
BoAHM~e , npe,~cTaDneHHa no~rpynn~ ~ . IIOMHrdO Toro ~~ AoTpe6yeM eke
~TO6~ axBizBaneHTHb~e rae~Ay coGoi, ~npeAcTaBneHHa ~ , axoA~u~ue B
1,1+ H TUB 3an~~c~BanHCb oAxHaxoBO - aTOro Tax ~e Bcer,~a ~o~xo A06HTb-
ca npocT~M HsMexeHHe~a 6a3HCOB. IIycTb rrpe~cTaB~:e~ii~e U,+(~~ rpynnx
'~ pacnaAaeTCA xa HenpHBO~x~s~-e npeAcTaBneH.ia {V~`~(k.~}, i.- i,~...,~;
~ Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
-34-
xpaT~iocTb npc,~cTaBneHirA V~~~(~~ B u+~~) r:in o6osxa~~in~ uepes~~,
a ero , pasraepxocTb - uepea s~ . B Taxora cny~ae v~HAexc ~ (= I, .... ,.N)
npH xoa~noaexTax Bei{TOpa ~ ~~~~~n 7 yAoGxo 6yueT aaNeHnTb cocTaBH~
i~H~cexcor.~ yes ( ~ i, .. , ~ ~ e= 1, , ~~ ~ S' 1 ~ . . , S~),
ui~ (~~ npvi 3TOA~f 06pa TY[TCA B
(4.32) u~ (k)=s?? S V``~(k).
es?JMt ~~ ~ em. St
a
~HallOrYi~iHO 3TO~y ecnii npeAcTaBneHite ~J'~ ~~~~ rpynnx ~ pacnaAaeT-
cn He fienpvieoAYl~~laie npeAcTaBnexHa V~ ~(k~ ~ ri,= 1, ..., .n(a ,npx~er~
~~~ ~'~~ TO DttAeCTO MHAeKCa rL
I{paTHOGTb V . (~G, Li T (~.) pBBxa Un~- ~
~
1~ .. , ~ cl.a ), Hyr~epy~u;ero KOltifnoHBHTI~ MaTPNi[ T ~(~, ,
,Mhl 6y,~era ynoTpe6nATb COCTaBHOLI YtHAeKC 11,u,a,:. ~ 1'1. = ~ 1 ~ ? ? ? 1. ~.7- ~
LL=1~. .~ U.ha', Q,= i,..., 5a); TorAa .'
(4.33) Tn'(ttw,mwi~~~=0nm.ouw'Va6w,(k,?; ~. - ~~
IIO~CTaBAAA (4.32) H (4.33) B (4.3I), 6e3 TpyAa nony~ae~a
4.34 _~c c 1~,~)
rAe
? mn. Uw,ee
rn u C 7t, wK ~1 J[I
OTC~Aa cne,AyeT,uTO d? uu
(4.36 ~T' `~ ~~) r(~
~~ts~~)-~~~~m~ue~miusl~~)
T. e. xa~:Aa.~ xol~~noxeHTa 7. ($) BexTOpa 7 (~~. pasJia raeTCA nvtmb no
d
Tee cTOn6uarl MaTp~iu T~"'(~) , r:oTOpne npi~HaAnez~aT Tory ~:e HenpHBO-
Am~o~y npekcraBnexHio rpynnn ~ , PTO x ~ -~ cson6eu ~aTpHu~
l~t~~, YI 3aHI2b11a1~T B H8M TA itie r4ecTO, PTO LI 3TOT ) -I~ cT0116eIj.
B CYlJIy (4.36) xOhinOHeHTbi ~~ `~J nOJIA ~ (x~ 6y~yT Y~Qe.Tb BYIA~
(4.37) ~~(x) _~,~~,';Ueu.?~ct(~)1?~~s~~>>
~ ~
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 rnslt0 ~I
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
-35-
HeMeAnCHHO nOny~I~eTCA M OdL'jafl (LOpRiyna AnA KOppenAuYIOHHOi1 MaTpY!-
rAe cyl~~aa 6epeTCA no Bcera ,uaa.:,~x nOBTOpr~I~IIII~MCA I~iH,i~CKCBM. HeTpyA-.F
1I cry ~ ~~.
HO npOBepYiTb, PTO c~yHKI;I-ILI l~l~i~s~~)T?~;y5 ~!~) Ha CaMOk! Aene dyer ~~
3aBVICeTb nHIIIb OT xnacca CI~e~HOCT1~i ~ IIO ~ , T.e. c~aKTYI~eCKYI
dy,ILy T cryllKuYIAL9YI oT x E x , axe. oT E ~ . 1rI3 (~. 37) H (4.35)
(4.38) ,~~Ci,x~) =~~uva6 u~~as~~l~Tvtius~~z~~s~uPvtw~~s'~~
rAe Cy1,~t~vlpoBaHHe Tax2Ce Be~eTCA TIO BCe~ ABa~::~~I IIOBTO~fIfOIgNMCA LIH-
c~, ~e~
Aexcala ig ~I ~uvac ~ - HeOTpNIjaT6J:bH0 OnpeAeneHHaA MaTpYiuB
170 BCeId CBO)rsluHOxxo>~ >aaT-
pJ/Ii~6i ~ (xt, xy, 6yAeT AaBaTbCA (~OpMynal.LI, nOnHOCTb10 aHaJiOr~~~HI~H
(4.37), (4.38) H (4.39) ; Tonbxo cyr~aavlpoBaHVIe no .a sAecb npxAeT-
CA 3aMeHYITb YIHTerpYlpOBaHYIeM COOTS@TCTByh~IgLIX 0%yHKuYIi~ ~~HO?TeCTBa
1
~ ~ ~ no Bce~ay npocTpaxcTBy ~1
B np>;~aexeHVlvl x KOHKpeTHbIM It~HOroo6p~.3IdAI,~ 11 H TLITIaM Ben>~t~KH
~T. e. npeACTaBneHLIAIri u(~ ~ ) 06IIjYIe ~ODldynbl MOI'yT, eCT@CTB6HH0,
Ccy1uCCTBeHHO ynpOCTI4TbCA. TBK,HanpLg,Iep,B cny~Iae none~I Ha cc~epe
X72 TpeXl:iepHOrO 3BKnvIAOBa npOCTpaHCTBa FZ3 CT81.jLI0HapH8A no~rpyn-
na ~G = Oz (noArpynna Bpa1~eHIrIH BOKpyr OCI~) OKa3bIBaeTCA xo1~e2~yTa-
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
::~-L a- ?~.-
~~~~~ ~~~
T1JiBH0~, TaK ~ITO BCe ee HenpLiBO,T;I4b4ble npeACTaBTieHYIA oAHOMEpHi~
(iiMeroT BYIJ~ V ~~~~~~_ e`"`Y , rAe ~ - yron noBOpoTa). HpoMe To-
ro B 3T01~ cny~ae Bo Bce HenpLIBO,~;IdIsSlle npe~cTaaneHVla rpynni~I ~=0a
nonHO~i rpynn~ BpalueHl~% R,3) xa::c,~oe npe;~cTaBnexHe V` _ rpynn~I
~2 BXOAIIT He donee, ~Ieki no p33y. II03TOMy 3,1~eCb BraeCTO 000TBB-
HF3X bIHAeKCOB ~QS , 1't'u,a, ~i T.A. 06~I~IHO r.40::yH0 OrpaHI~I~II~ITbCA np0-
CTbIlti9Li I~IH~BKCaMH ~, rti H T.~,. BOGnonb30BaBWI4Cb Ranee ~~opMynalex
AnA Bcex Ma.TpLi~iHb13C 3JIe2,1eHTOB T+~?,) npeAcTae.~IeHH~ rpyntl~ 03 ,
yKa3aHHbIrdYl B ~23~ , HeTpyAHO HanIdGaTb ABHI~l~1 BYIA COOTHOIIIeHi?YI
Hi~x I~ YI30TpOP..HI~IX nonel~ B rL-Mepxo~ 3BKT.YIAoBOra npocTpaxcTBe
_ Mw/
~-~~ On ( Mw - rpynna AByl~:exv~I~ FLw , On - rpynna n -
-MepH~x Bpai~exH~f) - aaAa~Ia ynpoi~aeTCa Teas,~TO BexTOpHOe n~ueA-
B Apyror.~ Ba WHOM ~raczxo~a cny~Iae - cny~Iae BeKTOpH~Ix oAHOpoA-
B6TCTBeHHO nO,ijOGpaHHbIM CneuvlanbHl,Rd (FyxxuYAM (CM.TBK?ne 3aMeT$y
[40, , B KOTOpO~I pa306paxa 3aAa~a 0 n0A06HOM ?ire HHBapYIaHTHO~G .
pa31f0:::eHYIH BBKTOpHbIX nOJIeI~ Ha "Cc~epe 5n (n+1)-2viepHOrO~ npOCT-
paHCTBB ~~,.
(4.37) H (4.38) AnA cny~Iaa, HanpHr~ep, BBKTOpHSIX OAHOpO,ZjHbiX cny-
L:aIiHbIX nOJIe~I Ha $z LI]Iy! TeH30pHbI}: OAHOpOAHbIX n01Ie~1 HeBbICOKOrO
paHra. Te ~Se pe3ynbTaTl~i MOu[HO nOJIy~IUITb, BOCnOJIb30BaBmHCb OnH-
CaHH~1M B [23~ LlxBaplrlaxTHbRa OTHOCLITeAbHO BpatgeHH~ pa3no~eHraeM
BexTOpHbIX In T8H30pHI~iX (He cny~al~xlax) none Ha crepe s2 rIO cooT-
cTaBneHise Ll rpynni~I Mn ABnReTCA B To ~e Bpe~a HenpHroA~I~e
yHVITapHI~a npeAcTaBnexl~eM noArpynn~r On , He 6onee paaa BxoAalq~-e
B xa~Aoe HenpuBOAiIMOe npeAcTaBneHHe Mw IIo3TOMy c~opa~y;~a (4.36)
sAecb npHHl~aeT BxA
(4.40) S(~) =J~T`5'~g~~mCda)
_,~,, ~Fq
~j ~ ~..l...,~ tv ri.x6i?:l A.8
Approved For Release 2009/07/09 :CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
rim ua a We. eL 4dUL. WaU.
38
We (op2Yiynbi Ann (x) (IInH Ann (!)) nOHHMaTb D TOM' CMbicne, qTO
(4.43)C`Q~ =J{x)~(x}dx,
X
rAe d.x HHBapHaHTF oTHOCIITenbHO npeo6pa3oBaHHLi e! Mepa 6X .
EAHHCTBeHHbIM OTJwI;Ieua 6yAeT TO, 1TO npH TaKOM. n0AXoAe K HSWHM
c Op3AyraL BbipareHIIA Ann ~x) (HnYI ~~, ) BnonHe MoryT.6NTb UI:.**
paCXOAAljmA1ICA - RHllb dU TOnLRO OHH CTaHOBLIJIHCL CXO,IjA1 HL,IHCA nOC
ne HHTerpHponal1Ha no Y(x)dx I'IJI.m te(g)dg II03TOMy yCJOBHA
CXOAHMOCTH (2.I3), (2.27),(B.I6) H T.n.AnA o6o0njexHbix none I. yze
He 6yAyT 06A3aTeJIbHERYH H gOJI CHU 6yAyT 6LITb 3aMeHeHIA Ha MeHee ze-
CTKHe ycnoBHA. ToqHblrl BHA 3THX ocna6neHHbIX yc)OBHII dyAeT onpeAe-
nATbCA aCHMnTOTHgCCKI?JJX CBONCTBSMH COOTBeTCTByI01jHX.M8TpX'IHLIX
3nCM8HTOB ILi H ct epH~IecKHx f yxxuHH B H8X6onee
BaY.HLIX KOHKpeTHEIX CnygaaX 3T14 aCHMIITOTHqecxme CBOIICTBa O6bIVHO
H3BCCTHLI Inx MoryT 6bITb 60 TpyAa fOnyLleHH, TOR qTO H3ytIeHHe
COOTBeTCTByI0u IX OAHOpOAHLLX o60614eHHbIX nOneH 3AeCb He CBA3aHO HH
C KORHMH HOBbIMH TPYAHOCTa1A1I (cp., Harrpi ep, p36OTbl [5] H [4~ , no-
CBAIjeHHbie obodljeHHw1 CnytIa2HLIM nOJAM B eBKJIHZOBbIX npOCTpaHCTBax
HOBbJe nOCTaHOBKH BO1 OCOB B03HIK6IOT, eCJH Hap1Ay c o6o61geH-
HbI1.IH nOnUUIH 3aA3HHbI1i,T.e.ocnadne-
HHe ycnoBHil peryJApxocTH, 30MeHfDIgMX (2.27), (3.16) H T.A.;{n3CC
Fir], RPIa 1 PItr n P V
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
I ; IsA r;. _..89. uktL I
06uKHOBeHHuX (He oOoOijeHHLIX) O,AHOpOAHbIX CnyVa9Hb1X honer ABJIAeT-
CA, C 3TOii TOLIKH 3pelilifl, nepeceqeHHe1 Ljenoro ceLericTBa OXBaTbZBaiW
ioig xx ,Apyr Apyra KnaCCOB o6odu exxblX CnygalnHbHX noneIrf,oTBetlaIoII1 x
Bce 6onee x 6onee Cya 4BaIOIAXMCA KnaCCaM ~yHXJAIU1 Henocpe) cT-
BeHHO 3a KnacCOM O6blxlJOBeHHbIX CnygaYIHbNX noneii B 3TOM ceiAer cTBe
6yACT cne,noBaTL KJIaCC OAHOpOAHLIX- cJIy1qariHbIx Mep - CnytlaMHIIX 4YHx-
gxHf g(5) I:1HO eCTBa S C X TOK14X,tITO EM(S)=E~Qs), E9(SS)9(SL) =
= E (~ S,) C9, SZ) npl nio6oi ~ E (no,izqepEHe1, BO H36eiaHHe
HeAopa3yMeHHA,gTO TepMHH "CnytIa Haf Mepa" 3Aecb HMeeT Apyrom
Ctt-1biCn) LIeM paHbMe B OT01 CTaTbe, rAe OH npHMeHAnCA JIHIUb K Cny-
1a3.HLIM ~,3HKIjHA1i1 MHozeCTB, npHHrMaIOI4HM Ha Henepecexaidlu#IXCA MHoxe-
CTBax Hexop.peJHpoBaHHble 3HalleHHA). HJiacc cnytlaNHHx Mep Mo&Ho
paCCMaTpIBaTb KaK KJIaCC 0606i(eHHLIX CnytlaYHbNX nonek,3aA6HHbIX Ha
BCeB03MOYHHX HenpepbuBHbIX C YHKIZHAX ; n03TOMy B OTJIH4He OT NO-.'
cneAyioiui4X KnaCCOB 0606tgeHHbIX nonell OH Mo-veT 6bITb onpeAeJleH Ha
nI06oM Tononormecxoi OAHOpOAHOZ1 npoCTpaHCTBe (T.e. rpynna of -
B 3TOM cnygae BOBCe He AOJZHa 06A3aTenbHo ABJIATbCR rpynno3 Ax).
B TOM. Ce cJbicne, qTO H AJIA o6o61lleHHbix OAHOpOAHbIX noneg
IrTo-renbC'aHAa HaA npOCTpaHCTBOM OCHOBHble pe3ynLTaTbl HBCTO 1-
uteri CTaTLX COXpaHAIOTCA H AJIA BCeX OCTanbHIJX KnaCCOB oco6njeHHUX,
noneli. OAHaxo BOnpoC 0 TOtIHOIA BHAe COOTBeTCTBy1OIAIIX "yCJIOBmg or-
P)
paHHgeHHOCTIl" (Hanarae1uX Ha Ko3C-c HIIHeHTbI V d Hnx Ha "onepa-
Topxy10 Mepy" 5(cL) ) AAA xakAoro 113 3TI'IX KnaCCOB Aon:ceH pelmaTh
cA oco6o.
(GL). IIOAA Co CII 1 HhII9H OAHOpOAHbimn -nDHDBIIeHHAMH. B TeOpJ4M
Cnytlai4HbIX npoL(eccoB HapaAy c npo1jecca1H CTaIZHOIIapHbnv1H XOpOIUO H.3y
qeHbi TaKie H. donee o6u a npoijeccu, HzselcuHe~ CTaWHOHapHble npxpa-
IgexHa (cm. Hanpxmep, [44] , [8] ). AHanori zHoe o6odi4eHHe IAOZeT 6UTb
rnn nrmipi mil V
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 a h 9"nr MID 1
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
npeAJIO>uexo x B OTHOLIeHHH nOHATI'IB oAHOpo.I;Horo cJyva.MHoro nOJA Ha
npOI4 BOnbxo} oAHOpO HOhA npocTpaHCTBe X . A mi exxo, none fi(x)
L1bI OyAehA Ha3bIBQTb nOJIehA CO cnyqaiIHbIAiH OAHOpO1lUTh1IY1 npupal4eHHA114,
ecihi BceB03I o .Hbie pa3HOCTLI g(xi)- g(xz) ~(xs,X2) 6YAyT npeA-
CTaBJIATb COOL! OAHOpoAHoe (OTHOCHTenbHO npe06p33QB8HHII (xi XI),
cnyqairHOe none Ha npocrpaxcTBe xxx
He cneAyeT CtHTaTb,gTO Teopm noneil c OJtHOpOAHbTMJI npxpalue-
HHAMLi HenocpeACTBeHHO CBOALITCA K TeopJ4H OAHOpOAHHIX nonei1 Ha HO-
BOM OAHOpOAHOM npOCTpaHCTBe: 3T0 He TBK, 1460 B npOCTpaHCTBe
X' X. rpynna yie He OyAeT TpaH3HTLIBHO11. Ho3TOMy B 06-
l
1geM Cny4ae ycTaxoaneHHe "cneKTpaJIIHHX pa3JIOZCHI1 " AJIA noneg c
OAHOpOAHbUAM nphipaIqeHYIAMH TpeOyeT nphlBnegeHHA HexoTOpblx HOBbIX C0-
o6pasceHV1g.
OCHOBH:1MH MHCJIeHHbUAH XapaKTepIHCTLIK6H}I HOJA C OAHOPOAHbIMM
npLIpaigeHYIAMH OyAyT nepBme H BTOpbie MoheHTbi pa3HOCTe3 9 (x1, xi)
(4.44) -~~ac,,x~,~=E(x,,x~,~~~(x~,x~,1x3,xY)~ Elx~.-x~)~?~3,xy)~
onpeAeAHTb 0619MiJ BHA :yHKIJHI . TO Ye KacaeTCA BTOPUX MOMeHTOB
(IyHKUMA L(xa,Z)nO:IHOCTbIO XapaKTepI43yIOTCA TeL1,gTO OHa yAOBneTBO-
pseT:,g)yHHLHOHanbHOMy ypaBHeHHm Yri,(x,,x~,)?m,(z~,,x3)= m.(x:,x,,)
H yCJOBLIIO fl( jxs,'gxi,)=?m(xi, xz,) ; OTCIOAa' O6LIqHO Yxie'' He TpyAHO
IgecTBeHHOe,TO B ChIny anreopSh4 eCxOrO TOxAeCTBO
,Tj (xi) xz 7,3, xy) , TO Bce OHM 6e3 TpyAa BuP8-U,@ DTCB '1epe3 -0B014
3Ha~IeHLiA .~(x,, x,, x3,x?npLI xy = JC1, ; eCJih none S (x) -- Be-
MLa MoHeM Aaxe orpallhP.UITbCA JMmb X3ygeHMem c yHKwI
(4.45)
3aBMMCauMX Been or AByx nepeieHHux. Kai 6LJIO Bb1ACHeHO McH6eprom
[45] Cl.IIKL(LiHi nOnHOCTbIO xapalTepm3yDTCB cneAytonii'tm CBOll-
,?5)* P". 101-7ri 9".'rl 5P nrNnn an
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
I wit Neu 8-uao:.-ym. cmrtr.. ovaar. ^
CTBOL; (POAdTBeHHI?1.i-'.CBO9CTBy nO2O}LHTe1ILHOi9 OnpeAeneIiHOCTH): jjp
JII061,IX rL, x1 ) ... )X.. C- 11 RI06LIX BenieC TPeHHbIX 'M X Lill,
TaliiiX, 9T0 a,z^ 0 , A0 H0 BbiUOJIH$ TbCS3 Hep B_BHCTBO 4
B KOMnaKTHOI cnyqae KJlacc nonei c 0AH0poAHHMH npHpan;emigmH coBne-'
neHHan ~yHHgHA Ha X (ci.EoxHep [II] ). OTCioAa cpa3y cJIeAyeT-,vTo
MHBapMaHTHa. (B cIviucne (3.I) ) n0n0ZMT.eJIbHO onpeAe-
xnacc TBHXX C YHIUU4Ji i3(xl, xz) COBnaAaer C KJIaCCON (YHKL(MM( B14Aa
Al (x, x)-.931(x1; x~~ , rAe x - npoi3BOJIUHaB ToiKa X , a
B cnyuae KO1naITHOrO np0CTpaHCTBQ X HeTpyAHO nOxa33Tb,gTO
L(UIrl (x,, x2) yAOBneTBOPOIN iIX (3.I) H (4.46).
Taxies o6pa3ol:4 onixcaHYle BceX (BeIueCTBeHHHHX) ?noneIl c omiopoAHin
nPMPaljeHIdBLIH B H3BeCTHOM c,LjHcJIe 3IBMB3JIeHTHO omicaxiiio Bcex (yHK-
(4.46) x.) GL, O.
j,~Ktl
AaeT C Kna0000M I1 OCTO OAHOpOAHbZX noJleL
MMa 6onee odigzx JIOKaJIHO KOMnaKTHbIX npOCTpaHCTB X noc ieA-
Hee yTBepxAeHHe yxe Oxa3MBaeTCa HeBepH ,a. ":"3xecb KiiaCC noneg C.
oAHOpOAHbiUIZ npxpan eHMAMH MOfCeT OKa38TBCB CyIueCTBeHHO mHpe xJlacca
oAHOpoAHbix none2I. Tax,B uaCTHOCTH, OOCTOHT Am B cnyxiae eBKnH-.
AoBa npoCTpaHCTBa (c rpynnoi'I TpaHC gM Hnu.o6i4eg rpyn
nOI ABM':eHM I B xaqecTBe rpynnbi CL1.nO 3TOMy noBOAy pa6oTy [4]
HUB C OAHOPOAHLimm npHpaIAeHHSII.IiI B ripoCTpaxCTBe Z,,, (oAHOMepHble K.
BexropHbie r: 1HoroI'depxbie) HrpaloT cyigecQQBeHHYIO pOnb B CTaTHCTHqecxOX1
TeopHH TypOyneHTHOCT;'i ; HIL".eHHO B 3T011 CBa311 BnepBble b1:Ino cCopMynH-
poBaHo x canto onpegeneHHe TBKHX noneMi (I{oJiuoropoB [46] ).. B elge
6onee VaCTHOM cnyqae X = Ri (T. e. AAR CJlyqallHLIX npOIJeCCOB ~(t) )
H3yqeH TaKZe H 6onee o6iamil xnacc npoiteccoB c I OAHOpoAHbuha1 npxpa-
IgeHHB>: H npoH3BOnbxoro nopiAxa ([4I] , [47] , [48) ) ; TCO Ha TaXXX
npoi eccoB Tait ce 1AO CeT 6bITb nepexeceHa H Ha cnyvai%I nonell Ha HeKO-
b nn CFI '~'~ C'19A
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
TOpb1X OTJm IHLIX OT fZi 0S1 HOpOAHbUX npoCTpaHCTBaX.
B cnyqae, xorAa rpynna & aBxneTCa rpynnog .hiz, Y O HO
pacchiaTpIIBaTb Cpa3y O6odueHHLJe cnyurnibie nova c OAH0pOAHUM14 np14-
p@AeHXBMl4. HeTpyAHO nOHBTJ , qTO Tame none L O:'TO OnpeAeJIHTb,KaX
none ygOBneTBOpaioIJ e COOTHOLleHMaM (4.4I) (wiui (4.42) ),
HO 3a18HHbie nXWL Ha nOAnpOCTpaHCTBe 4 C yHIC.14 1 I'I3 TaKI4X,g1TO
(4.47) J~P(x)dx= 0
(cp. {4}). OTO onpeAeneHHe ,onyOxaeT ABJIbHeiiliHe O.000Ig6HI14 CBa-
3aHHble C p8CC 4OTpCHHe1J OAHOpOAHHIX CJIyqa:iHHX nOnek Ha HOKOTOpbiX
Apyri'ix nl'IHeI HNX IOAHPOCTpaHCTBaX npOCTplHCTBa ; B X1aCTH01 ::C1iy?
qae noneI Ha npaMorii R . Ha 3TOIJ nyTH .THKI:{e IJO?{HO,nOCTpOHTB .TeO-
p'iio noneii c oAHopo,HLBJH npHpauiernaIJH f.-ro nopJAxa.
Approved For Release 2009/07/09: CIA-RDP8OT00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
~UR UJ* Waj-iL ",L
MEPATYPA
I.I.J.Schoenberg,"Metric spaces and completely monotone fun-
ctions" Annals of Math.,vol.39 (1938),PP.811-841.
2. A.M.firJIOM, "OAHopoAHaa x x3OTpofHa.E Typ6yJIeHTHOCTb B B$I3KOA
cwxMaeMO Ei KOCTx"., 143BecTxa AxaA. HayK CCCP, cep.
reorpac. x reoclix3., T.12 /1948/, CTP. 50I-522.
3.J.E.Moyal, "The spectra of turbulence in a compressible
fluid; eddy turbulence and random noise",Proc.Cambr.
Philos.Soc.,vol.48 (1952),pp.329-344.
4. A.M.firJIOM, "HexoTopHe xxaccn cJIytlai HHx nonei B TL-MepHoM
npocTpaHCTBe, poAcTBeHHNe cTaLlnOHapHuM cJ1y-Iat HHM
npoueccaM", Teopxsl BeposiTH. x ee npxMexeHxx, T.2
/I957/ cT. 292-338.
5. "Isotropic random current",Proc.3-d Berkeley Sympo-
K.Ito,
sium on Math.Stat and probab.,Berkeley and LosAn-
geles,vol.2(1956),PP.125-132? 0
6. A.rd.0(IyxoB, "CTaTxcTxgecKx oAHOpOAHHe amynaliHxe nOJU Ha ct)e
pe", Ycnexx MaTeM. Hayx, T.2, 16 2 /1947/, cTp. 196-
198.
7.I.J4Schoenberg, "Positive definite functions on spheres",
Duke Math.Journal,vol'.9(1942),pp.96-lo8.
8.J.L.Doob, "Stochastic processesf,New York,1953.
9.L.Pontrjagin,"Topological groups",Princeton,1939.
-np
' ,7 }~ ~j 77 y 1 p.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
7
'+LLi,r l I
44'
I0. A.Weil, "L'integration dans lea groupes topologiques et
sea applications",Paris,1940.'
II. S.Bochner, "Hibbert distances and positive definite
functions",Annals of Math.,vol.42(1941),pp?647-
656,
12.K.Karhunen, "Veber lineare Methoden in der Wahrscheinlich-
keit%rechnung", Annals Acad. ' Sci.Fennicae,A,No.37
(1947),PP.3-79?
13.H.Cramer, "A contribution to the theory of stochastic
processes", Proc.2-nd Berkeley Symp.on Mathew.
Stat.and Probab.,Berkeley and Los Axngeles,1951,
pp-329-339-
14. A.A.PaiaxoB, "rapMoHHIecxx2 axaxxs Ha xoruhyTaT1BHxx rpynnax
c Mepo;i Xaapa a TeOPHH xapaxTepoB", Tpy MaTeMaT,
HHCTHTyTa Hrreax B.A.CTe=Ba, A I4 ./I945/, 86 cTp,
15.J.Kamp4 de Ferriet, "Analyse harmonique des fonctions a14-
atoires stationnaires d'otdre 2 sur un groupe
ab4lien localement compact",C.R.Acad.Sci.,Paris,
vol.226 (1948),pp.868-870.
1ITh6orie des fonctions alSatoir4
16.
Fortet
R
,
.
Aalanc-Lap ierre,
Paris,1953.
17. M. M.I'eaibc A. A. Pa &i oB, "HeflpMBO He yxnTapHHe npeRcTae
.neHxx aioxa II Ho dmoMnaxTHxx rpyfn", MaTeM. cdopHBK, j
T. 13 /55/ /1943/, CTP. 301-316.
18. M. A. Hakmapx, "HopMxpoBaHHxe xo mL a" ,MocxBa, 1956,
I9. Harish-Chandra, "Representations of a semisimple Lie
group on a Banach space",Trans.Amer.Math.Soc.,
vol.75 (1953),PP%185-243.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
i Ui w x6,K
RILY
45.-
20. J..Dixinier, " Sur lee representations unitaires des
groupes de Lie alg6briques",Ann.Inst.Fourier,
vol.7 (1957),PP?315-328.
21. G.W.Mackey,"Borel structure in groups and:, their duals",
Trans.Amer.Math.Soc.,vo1.85 (1957),pp.134-165.
22. A.Guichardet, "Sur une probl&me pose par G.W.Mackey",
C.R.Acad.Sci.,Paris,vol.250 (1960),pp..962-963.,
23. H. M. rex4aHA, P. A. M moc , 3. H. Manxpo, "1IpeACTaBJteHxx rpyn-
ns BP=eHMH x rpynmi Jlopeima",MocKBa, I958.
24. H. Fi. BxxexxxH, "Becce.neBN tIjHxuxx n npeAcTaB.nemU rpynri
eBKJIHAoBHX ABxDeH1Nt" , Ycrexz MaTer. Hayx, T.II
/I956/, cTp. 69-II2.
25. V.Bargman," Irreducible unitary representations of the Lo
rentz.group",Annals of Math.,vo].Q8 (1947),
pp.568-640.
26. H. M. re. c)aHA, Psi. A. Ha mapx, "YHxTapHHe npeACTaBJteHBA x nac-
cx-qecxxx rpynn.," TpyAa MaTeMaT. RHCTHTyTa BMeHB
B.A.CTelJtoBa, A 36, /1950/, 288 cTp.
27. H.R.BxxeHxxH, 3. JI. Axxta, A. A. JIeBHH "MaTpxime aJieMeHTH HeIIp%,
BOA1Mxx yHxTapHHx.npeAcTaBxexxA rpynnx eBx uo
BUX ABxZeH11A TpexMepxoro npocTpaHCTBa x Bx
cBOIcTBa", AoxJiaAH AraA. Hayx CCCP, T. 112
/I957/, CTP. 987-989.
28. H.R.BBJIexxxH, "MaTpWIHHe aJieMeHTH HenpxBogHMEIX ylplTapuHx
npeACTaBJteHxl%i rpynriu BeIgecTBeHHux opTOroHalmHux
rdaTpxA i rpynnx ABxxeHHA (t- i) Mepxoro eBxmauo?
Ba npoCTpaHCTBa",AoxataAH AK i.Hayx CCCP,T.II3
/1957/, cTp. I6-I9.
LP I
~+ 7
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FW( U jLj6S'%L L JNLI
46.-
29. H.a.B IeHxIIH, "MaTpwrHHe aJIeMeHTEI HeIIpHBOAHMHx yHHTapHax
IIpeAcTaBJreHxR- rpynri ABH eHHi IIpOCTpaHCTBa JIo6a-
tleBcxoro z ododueH we npeodpasoBaHxa oxa McJiepa"
Aoxiraru AxaI. Hayx CCCP, T. II8/I958/, cTp. 2I9-
222.
30.E.Cartan,"Sur la d4fermination d'un syst4me orthogonal
complet dans un espace de Riemann symmCtrique
clos",Rend.Circ.Mat.Palermo,vo153 (1929),pp.217-
3I.H.Weyl, "Harmonics on homogeneous manifolds", 252:
Annals of Math.,vol.35 (1934),pp.486-494.
32.A.Erd4lyi, "Higher transcendental functions",vol.I-III,
New York,1953-1955.
33. (D.A.Bepe3XH, H.M.re,m wig, "HecxoJlbxo 3aMeiaHH r x Teoprax
c piieci x (]THXIIQ Ha cx=eTpTaqeci x p1MaHOBHX
MHoroodpa3xRc", ' TpyAN MocxoBcxoro MaTeM.O6UL-Ba,
T. 5 /I956/, 3I2-35I.
34. M.M. reJrbc al=, "Cc epmeci e c ryHxumm Ha cmweTpx4ecic x px-
.ti
MaHOBMX IIpocTpaHCTBax", "AoKJISAH AxaA. Hayx CCCP,
T.170 /I950/, cTp. 5-8.
35. M. r.Kpek!H, "3pMI TOBO-IIOJIO: XTeJILHNe H pa Ha o,AHOpoAH
IIpoCTpaHCTBax", 'q. I-II, YxpaHHCKHia MaTeM.tKypxaJI,
T. I, ! 4 /I949/,cTp. 64-98; T. 2 I/I950/,
CTP. I0-59.
36. 14. h1. re.I a , M. A. HaiiMapx, "YHHTapHHe npeAcTaBJreHgx rpynnw
JIopeHua",H3BecTnx AxaA.Hayx CCCP,cep. MaTeM.,
T. II/I947/,cTp. 4II-504.
;r
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR OFFlCIP i USE O "Y
47.-
37. 9?. A. BepesHH, c. K.Kapne.neBxq, "3oxa nbxxe cc epxiecxxe c ryxxua:
x onepaTOpn JIannaca xa HeKOTOPLM cxMMeTpW ecxxx
npocTpaBCTBaX", AoxaiaAn Axau.xayx CCCP, T.118
/I958/, cTp. 9-I2.
38. A. H. EoraeBCKxH, "Bb1gxcjiexlae 3OHa nLH13X cc epxiqecxrdx (Dyma "!
Aox na,Au AxaA. xayx CCCP9 T. 129/I959/, cTp. 484-
487.
39. IO.A.PosaHOB, "CnexTpa.MHaa TeopIa MxoroMepHbat cTauxoxap-
JINX npoueccoB C AKCKPeTHNM BpeMexeM", Ycnex7a MaT
HayK, T.I3, /I958/, cTp. 93-I42.
40. A. A. KI pxJ.noB, " HPeAcTaanex1x rpynn Bpautexxia 71-Mepxoro
eBK AoBa npoCTpaHCTBa cc epWIeCK BexTopaMx
n0imn",Aoxna W AKaA.xayx CCCP, T. II6/I957/,
CTp.538-54I._.
4I. K.Ito, "Stationary random distributions", Mem.College Sci.
Univ.Kyoto,Ser.A vol.28 (1953),pp.209-223.
42. I4.M.rezbc axu, "06o6utexxxe cnynaiaxxe npouecca", Aoxnajw
Axa,R. Hayx CCCP, T. I00/I955/, cTp. 853-856.
43. H.M.re II axu, r.E.IlhnoB, "RpOCTpaHCTBa OCHOBHNX x o6oftes-
ix c rxxuxg" /" 06o6uieHF, a cpyxxr. ax" , Bbtn. 2/, Mocxea
19580
44. A.H.KonMoropoB, "KpxBBe B r 6epTOBCxoM npOCTpaHCTBe,
xHBapgaHTHBe no OTHOMeH= x oAxonapa1eTpxiecxo2
rpynne AA cexx#" , AorZagu Axau. xayK CCCP, T.26
/1940/, cTp. 6-9.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR OF f lCIet L SSE ONLY
48.-
45. I.J.Schoenberg, "Metric spaces and positive definite
functions",Trans.Amer.Math.Soc.,vo1.44 (1938)
pp?522-536?
46. A.H.Kojworopoe, "Jloxanbxa.a CTpyxTypa Typ6yneHTHOcTH B Hec-
HMaeMOH H KOCTH up14 o'qeim 6o.Ib1Hx g iwiax
Pe iHo nbAca" , Aox. aaAN Axa ii. Hayx. CCCP, T. 30
/I94I/, CTP. 299-303.
47. A.M.$ rJtoM, "Koppens1OHHaa Teopaa npoueccoe co c y akx1Mg
cTat~uxoxapxnM TL-Mx nplp=eHH Z", MaTewaT. '
c6opirx, T. 37 /79/, /I955/, CTp.'I4I-I96,
48. M.C. Mmcxep, "Teopxx xp1Ebnc B rzim6epTOBOM npOCTpaHCTee
co CTauHOHapH>3MH n,-Mx npMpaigeHHxMH",}I3eeCTHa
AxaA. Hayx CCCP, cep. MaTeM.T.I9/I955/,3I9-
3450
n -nR
yJ~ "': Ii L tlJ U v?i L ~' V sa
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
121
, ~ nee :,~~ RS~~ ~i-99H. r
ON THE PROBABILITY OF LARGE DEVIATIONS FOR
THE SUMS OF INDEPENDENT VARIABLES
Yu. V. Linnik
USSR Academy of Sciences
1. Introduction
ti
The classical theory of the summation of independent
random variables as expounded in the book [y~] in its simplest
case considers the increasing sums Sn = X1 + ??? + X. For
the properly normed and centered sums Zn Sn/Bn - An the
behavior in the limit of the probability measures generated
by (Zn) on the real axis is studied.
The most general theorems are the integral1 on the
limit behavior of
(1.1) P(Zn ` x).
Although the theory of local limit theorems is rather well
developed (s), it is not yet of such finished character as
that of integral limit theorems. Limit theorems for the
expression (1.1) usually suppose that n -- co and x is a
fixed number.
However, many problems occurring in such different
fields as mathematical statistics[]n1 [ information
theory statistical physics of polymers
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR
rubber. chemistry (17], and even analytical arithmetical
require certain information on the limit behavior of (1.1)
not contained in the classical limit theorems. The informa-
tion required concerns the asymptotic behavior of
(1.2)
P(Zn > x).
for "large values" of x, that is for x = xn increasing as
n increases; the corresponding problems will be called
o the rcbability '
problems on ill! of large deviations. As
probabilities of events of this kind are small,?' the
usual methods of establishing the limit theorems (dharacter-
istic functions, partial" differential equations) are too
rough to give satisfactorily general results and the desired
asymptotic results were considered in the literature under
certain very stringent conditions imposed upon the variables
Xi .
The first theorem on probability of large deviations
K- A
was published by OWN Khintchin# [101 in 1929 and related to
. u .
the particular case of IM Bernoulli variables. The same
case was treated-more completely by W=ft A Smirnov (16]. In
N ,
1938 appeared the fundamental paper [4+] of lA Cramer contain-
ing the first result of a general naturein the theory of
v,1. r
g deviations. It was improved by Feller [] and
U.V. V
Petrov (12) .
No1.r, tie shall formulate Petrov,s result, restricting
ourselves to the case of identically distributed variables
for the sake of simplicity. Let X1, 2 ..., Xn be a
~ ~r~^ . y n.f1r:M 4~M.R 7
r
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Sn=X1+ +Xn, Zn , cr-VE
x
au.
P(Z < x) , G(x) = ___
n ~r 2fi -co e
~Ly
FOR 0a"u~ ,
sequence of independent identically distributed variables with
and
must hold for some a > 0. Then for x x = o(/ ),
Cramerts condition
(1.k) 4 E exp (aIXjI).
n -- oo we have
(1.5)
(1.6
11?
x 1+0
`-X-)
n
Vn
- 'i n
where T(z) is a power s?riees involving A cumulant.s of the
variables X and convergent for tz.( -`- E0, eo '' 0?
t ltd ] , 5 introduced s stemat i
Later on, Richter~(l3], 11 y
cally the saddlepoint method into the theory of
deviations.' For a particular case this was done earlier by
Daniels (h. Under Cramerts condition (C), Richter
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
1 - Fn(x) exp r 3 x
- i -- ?
1 G(x)._._.1
n
Sn
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Foil
aL' Jam. u~ a~ Z ,1
~t
deduced several local limit theorems for tleElft deviations,
established the connection of Cramerts method with the
saddlepoint method, and investigated the necessity of
condition (C) for the formulas (1.5) and (1.6) to hold for
X = O( V7).
All the results hitherto obtained used Cramerts
condition (C). The analytical meaning of the condition (C)
is that the characteristic function (ch.f.) of the X is
analytical in some neighborhood of zero, and so in the
corresponding strip. This.enables us to apply complex
function theory and the saddlepoint method.
But if the condition (C) is violated, the methods
hitherto applied fail. The purpose of this paper is to give
some applications of a new approach which enables us to
obtain rather general results. Of the class of problems
subject to this method we shall treat here only the problem
of tft normal convergence and the problem of limit theorems
valid for all values of x for n
2. Zone of normal convey ence: into real limit theorems
We consider here the normal convergence problem for
deviations for the sake of simplicity only for inde-
n
pendent identically distributed variables.
arsfe
Let X
1, 2, ???, Xn, ??? be independent identically
1,
distributed variables with E(X.) = 0., D(X.)
J
Zn _(X1 + 2 + ??? + Xn)/ . Let T(n) -1 00 be any monotone
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
function. The sequence of the segments [0, '!(n)] will be
called Zhe zone of normal convergence (z.n.c.) if, for
n -4 a*,
(2.1
_ r ?? a-u2/2 du
J x
A
for any x e (0, T z.n.c.. [-T(n), 01 is defined
similarly. The definition does not require the convergence
to be uniform, although in all the theorems obtained it will
be.
Under Cramer's condition (C) for any '(n) = a(n1/6),
both [0, !(n) ] and [-T(n.), 0] will be z.n.c.- The zones
with T(n) = o(n1,/6) will be called narrow.zones. Our.two
first theorems relate to the zones with T(n) = nn, where
a > 0 is a constant.
Theorem 1. If for-an y a %-1/2, the zone [0, na] and
the zone [-nU, 0] are z.n.c. then all the variables Xi
P[Zn > x]
-6(e
Of course, this is also sufficient for zones
[0, na) and [-na, 0] to be z.n.c. This last fact is
trivial. Thus, we see that it is sufficient to investigate
the values of a < 1/2.
are normal.
- be a monotone function
Theorem 2. Let p(n)
increasing as slowly as we please and let 0 < a < 1/2. If
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
GNL
a 1/6, t necessary, condition for W. zones [0,
4a
(2.2)
E exp IXj( 2a+1 < 0.
nap(R)]
+he
This condition is sufficient for, zones- (0, na/p(n)] and
[- na/p(n), 0] to be z.n.c. and the convergence is then
uniform. If 1/6 ` a 4 1/2, consider the _sequence of the
critical numbers
(2
3)
Let
(2.k)
1 1 n. .., 1 ss++ 1... _.~ 1-.
4' 10 2 s + 3 - 2
ls?l
2 s w+ 3
1 ss ' 2
a .
2s'+4
If the zones (0, nap(n)]' and [-nap(n), 0] are z.n.c._,
the condition (2.2) must hold and moreover all the moments
of X 3 up to (s + 3)iust coincide with the moments of the
normal law. These two conditions are sufficient for the
zones [0, na/p(n)] and na/p(n), 0] to be z.n.c. This
convergence is then uniform.
As the normal law is completely determined by the sequence
of the corresponding moments, theorem 1 is an immediate
consequence of theorem 2.
We consider now the narrow zones with `Y(n) = o(n1/6)
other than [Q, na]. ZiL_ condition necessary for the zones
(0, 'Y(n)p(n)] and [-T(n)p(n), 0] and sufficient for the
zones [0, T(n)/p(n)) and E- '1(n)/p(n), o] to be z.n.c. is
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4 \
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
where h(x) is a monotone function depending upon T(n).
It is simpler, however, to describe T(n) in terms of
h(x). To this we consider several classes of functions
of h(x). The functions h(x) will be assumed to be
positive, monotone, and differentiable.y
Class I will denote the functions h(x) ~rf the
aarw
of the type
(2.5) E exp h(IX~I)
condition
L
"0 .
2.6 ) ( ~cf x) 2+C0{~ h(x) xl/0
, x
Here 0 > 0 is any small fixed number. Functions increas-
ing faster than .x1/2 will not be required for the narrow
zone investigations.
Class i consists of the functions h(x) under the
condition
(2.7)
&Y 110~ 2+C .
Pi(x) A x h(x) ( x) 0,
where x > 1, p1(x), p2(x), ??? in what follows are given
positive monotone functions increasing as slowly as we please.
Class I consists of functions h(x) such that
(2.8) 3n x 3 is a gives censtant, The inequality
h(x) > 3 Pi x is connected with the existence of the third
Consider now the functions of 9'lass I as defined by
moment.
(2.6). We put
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
011
(2.9) h(x) = exp [H (fix)).
Then H(z) is a monotone differentiable function. We
introduce the following supplementary conditions
(2.10) H!.( Z) ` li
(2.11) H' (z) expLH(z)] -- CO
These conditions follow from (2.6) if H'(z)
mine new functions A(n) by means of the equation
is assumed to
be monotone; otherwise we adopt them to simplify the results.
Given a function h(x) of class under the supplemen-
tary conditions (2.10) and (2.11) or of class we deter-
(2.12)
h(-,/_h A(n)] = [A(n)32.
Theorem 3. The condition
<
(2.13) E exp MIX i D"
where h(x) belongs to class [with (2.10) and (2.11)] or
ten, o
class is necessary for the zones [0, A(n),p(n)) and
t"3
[-A(n)p(n), 0) to be z.n.c. and sufficient for the zones
[0, A(n)/p(n)] and [-- A(n)/p(n), 0] to be z.n.c. The
convergence in this case is uniform.
We pass now to the functions h(x) belonging to
class III. This case can be studied by classical means [].
For the sake of completeness we formulate
Theorem 4. Condition (2.13), where h(x) belongs to
class is necessary for the zones
and
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 i Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
to be z.n.c. and sufficient for the zones
J 7
L_ 0,01ry)?1
nt-n
/p(n) ) and r _'V-
- P to be z.n.c.
Thus, if E J X ~ I lY` _ CO for a fixed M, the 'z . n . c . cannot
be essentially wider than [0, y4]. It is, roughly
speaking, of this size if EjXj1M 4 oo with M > 3. The
case EIXjI3 =_00 (nonexistence of the third moment) was
studied by several authors (see [] for the literature).
The case (2.8), which is class III, of the functions
h(x), corresponds to slowly decreasing "probability tails"
P > x). In this case, as we shall show later, a new type
of limit theorems holds: limit theorems valid for the whole
x-axis.
valued variables. We can consider also probability
for variables possessing a probability density or for
In the preceding section we considered integral limit
theorems. We pass now to the local limit theorems relating
to normal convergence. These theorems are usually considered
Local limit theorems
measures on the ring of the'i i-egral of an algebraic
number field. We shall restrict ourselves to the class (d)
of all random variables possessing a continuous bounded
density g(x).- Then Zn (see section 2) will also have a
continuous density p' '(x). The zone [0, T(n)] will be
0. Zn
called.t~h-- zone of uniform local normal convergence
(z.u.l.n.c.) if
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
(3.1)
10
as n ---t oo, uniformly for x c to, T(n)). The z.u.l.n.c.
(-Vn), 01 are defined similarly.
Theor .5. For the variables Xi belonging to the class
(d) the z.u.l.n.c. behave with respect to the necessary and
n
'fhe local limit theorems for A deviations are
easier to prove than the corresponding integral ones, by the
-b
theorems 1 ZIL
sufficient conditions indicated in theorems l44 in the same
way as the z.n.c. for the general random variables in
jr,
method proposed here.
the existence of the probability
density g(x) greatly facilitates the proof.
i
We shall be able to expound here the proofs for only the
simplest cases so as to present the new approach in its most
-to
transparent form; the proofs of all the theorems 1A5, although
not basically different, are more involved and will be
published elsewhere. In particular, we. shall treat only the
zones to, na,] and (-na, 0) with 0 < a < 1/2 and only
local limit theorems. However, since a part of the necessary
conditions for the integral theorems is almost trivial, we
shall begin by dwelling upon it.
.i :/.,.,. ~;/J'.:I:~ it
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
FY~~Widd:r kl lli~ ~+~G 1LY
11
Let the zones [0, nap(n)] and [-nap(n), 0] be z.n.c.
We shall prove that
(4.1) E exp
Suppose (4.1) does not hold. Then it is easy to see that
a
there exists either .rhe sequence xm --3 oo such that
(4.2) P(X1 > xm) > exp
a
or. ;race sequence -xm -) -oo such that
4a
(4.3) P(X1 -x1) > exp '2 2xmc~1
Suppose that (4.2) holds. For a. sufficiently large m,.
choose n such that xm = na +1/2p(n). The zone [0, nap(n))
being .Oie z.n.c., we mint have
n--p(n)n --'Lp(n) J
(4.4) P-zn > 2-- ... = 16
But the event Z > np(n)/2 will surely occur if the two
n
independent events X1 > na +1/2p(n) and
I(X2 + X3 + ??? + Xn)/ vI- 0 is a constant,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 1~
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
na +1/2p(n)
4a
12
2a
c 0 exp -2n ) 2a+1
?
w
by (4.2). .kS a < 1/2, Zkwe 4a/(2a+1) < 1 and (4.5)
contradicts (4.4). The case (4.3) is treated similarly.
The proof of the necessity of (4.1) for the. zones
[0, n p(n)] and [-n' p(n), 0] to be z.u.l.n.c. is
constructed in a similar way.
We now pass to' #rocai. limit, theorem. Let the variables
X possess the bounded continuous density g(x). We
introduce some notation. By the letter B we shall denote
a bounded function of the parameters considered, not always
the some. 8p, $1,????; P0~ E1, ?'- will be small
positive constants; CO, Cl" (;l co: cl,??? positive
w
constants. -Denete-
(5.1) cP(
du.
The function I cp(t) 12 is a nonneg/atiivee Fourier transform W&
and therefore (compare [1], p. 20) 1g(t)12. a Ll(-oo, co),
so that
(5.2) fCO lm
A
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
itxj = I itu
1 dt < ~.
2
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
13
Hence we have
(5.3)
itx dt.
J - [,P(t) ]n e-
-Vn
~
Let (k.1) be Suppose first that a 1/6. We
must prove that [0, na/p(n)] and [- na/p(n)
0] are
z.u.l.n.c. We shall study only the first zone, the second
one being treated analogously. Take x such that
(5.k) 0 { x < as
p(n)
In view of (1.1) the function p(t) is infinitely differen-
tiable on the whole axis and so, for any T > 0, It+ g To
and p > 0 an integer,
(5.5) `4`t) = qP(0) + ?(o) + ...,
It
1
(P l).t PT Rp(t)
where
(5.6)
Moreover, '.91(o) = 0;
E0,
. X V4""A
-1. From this// for
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
2
(5.7) 1(t) = 1 - 2 + Bt3;
Ip(t)I e
(5.8) I (P(t) I < .1; cp(t) --' 0
0'
as t -+ ? oo.
(5.3)., and (5.8) we 'getT
(5.9) PZ (X) = '` n J eo (cp(t) ]n a .-~!n' itx dt +B e -con.
n 2r 0
Combining (5.2),
1
(5.10) ? = 2 - a
In view of (5.7)) for n-?
(5.11) IcP(t) In ` 1 -
obin
Itl { e0, we A
1-2?)
= B.exp(-c1n
= B exp(-c1n
Hence from (5.9) we
(5.12) pZ (x) =
nu?
r _ (cP(t) ]n
V_~ ?
+Bexp
6.
The function (5.5) is not analytic in general and so
-he Taylor series for it diverges. We must Choose an
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
/ab~ lu ?w~
15
appropriate approximation to it in the segment Iti
that is, choose a convenient p in the formula (5.5). We
need estimates of cp(1q)(0) q < p. We get
(6.1) cp(q)(t) = f CO eitx (ix)q g(x) dx.
CO
Hence
(6.2) I Cp (q) (t) I .< f 1x18 g(x) dx.
Putting k = (1+2a)/ka, we obtain from (k.1)
(6.3) f 00 eXP((xIl/k) g(x
CO
dx < co)
4iltence we easily obtain
(6.k) I Cp (q) (t) I = Bgr(kq) .
ur~ to - ?
K(.0) = 0,
Let us deaei% K(t) _ T(t)
From (5.12) we ge_t-
(6.5) PZn(
n
e tnK t) - fin` itxl dt(+
n
+ B. exp (-60n )..
Moreover, from (5.7) we conclude that, for Itl 0. Let now
For 3 r s Cl,, when C1t is any constant, we have for (7.7)
the value
(7.9) B e x2/2 ~p - r
We are thus led to
can be subjected to the
iex)d9
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
irwAL t W L1
21
Summing these for r = 3,.4,..?Cl we get the estimate
(7.10) B e-.' 2/2 1
[p(n)J3
Consider now JJi < r _< m. We use the following expression
for Hermite's polynomials Hq(x)(see [7 p. 193):
~[q/2J (-1)s(2x)q-2s
(7.11) Hq(x) = q. E
s=0 s. (q, ,:. 2s)!
X 04'
(7.14)
= Bng? = B exp (q?~,Q n):
(7.13) -p^ q-p p (1- 2p) nq - (1-2p)AL e'
V V
Multiplying (7.13) by n-q/2 = exp [-(qa)/2J and by
B max exp q+ B + (1 - ap) `~a n - ~~, p (n )]'
10 /Oft 1 (~ - x
Then (7.12) has the value
Hence (x 0'--; Oo assumed to be equal to 1),
xq-2s
(7.12) lH (0)(x)I = Bqq! max
s=q/2 s: (q - 2s)'
Let s = qp,0 < p < 1/4
x < na/p(n) 1 < q m.
A 0
(7.15) B max exp q[B - (1 - 2p) Yid p(n) + p.n.& q - p2a
/Ie get, after an elementary computation,
by
As ~2n q < -,Z )m = 2a Wn - ,fin) pl(n) + B, we can replace (7.15)
(7.16) B max exp q( B - (1 - 2p.) -, ,- p(n) - p Q.n p(n) )
If p 1/k, p aril p(n) > i/ p(n)/k ; if p
with a sufficiently small n, and proceeding as in section 7, we
obtain after
some computations similar to that of sectJon 7j
0Z n (X) = -:~ I ~2r
(q,c')
for a sufficiently small n > 0. Taking Ainto account
we see that the second term on the right side is of the
form
(9.8)
s0+3E
as n - co, where a?0 0 0. Hence, there is no local normal
convergence in the'zone [0, n1/2 -43 and, likewise in the
zone [0, nap(n)].
L
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
25
Hence, if there is .e uniform local normal convergence
in the zones (0, na)N[-na, 01 for all a < 1/2, then all Tr
must vanish and so all the X3!s are normal.
io. /LM Q,to
There is an interesting class of probability densities
g(x) for which an integral limit theorem for the normal sum
Zn holds for the whole x-axis.
Consider the class of all event/ continuous probability
densities g(x) such that for x a. 1
(10.1) P (X1 > x)= ~g(u)du = Aa + A-p +...+ Ada+S
f x a x
x
X a+5+e
where a z 3, a being an integer, Ai being constants. Let
X1, X2,..., Xn be random variables with 44~ probability
density g(x) of this class. ' E(X~) = 0. Daneve W
? D(X) = 2, z (x1
;/-
n6-n
Theorem 6. For x 1, and as'n --' co we have uniformly
with respect to
PtZn > x}
1 e -u2/2du + r(x, -4W -r fx
where r(x, N n') is a rational function of both variables
determined by the coefficients A3,..., Aka+S in (10.1).
Moreover, for x > n3/2 + 1/a 'e fl `n we haves-
e evenness condition is assumed kor simplicity otgy:
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
f~.
ii. LY 26
yI Hk,al,...
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
(10.3) r(x, i/n) ti nP(X1 > a x'V-n} n f g(u)du.
00
-0
Of course, g(x) being even, an analogous relation holds for
x ~-l, while for ' 1 < x < 1 the classical normal convergence
that D(Xj) =--11 we have [for x > 1, n -->
theorem acts.
Simple examples of theorem 6 are given by rational
densities. For instance, if g(x) = 2 7r(x2 + 1)23 , so
P(Z.n > x)
1 du + 2
' 100e_u2/2
7r V -n
x 3
nP{Xl > x V n}
(10.5) 2 x3
37'll n'
We shall indicate here briefly the principal points of
tor x
the corresponding proof. We take a = 1. The case
(10.6)
n3/2 + 1/a-ri
A
is treated in an elementary manner. If y > n, then the event,
Sn > y implies at least one of the events Xi > y/n. Denote
by Y,'fmr k < n, the hypothesis that
Hk
,a112
,.,..'):ak
Xal > y/n, X, > yin,..., Xa > y/r~while this is not true
for any other Xj. Hence
F1 Pl Sn > y} = Z P(. H1 a ) P (Sn > y I Hl a
(al) 1 '
+z Z P{ Hk,a1, ...)akJ P Sn
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
27
IKY
the summation being extended to all ordered sets of distinct
numbers al < a2 , y.) ti (Z al) P(Hl,ai) P (Sn > y I Hl,al)
= nP(H1,1) P (Sn >' y I H4} .
Moreover, it is easy to see that
(10.9) P( Sn > y I Ham} P (Xl > y I X1 > n- } ..
Inserting this into (10.8) we get:
2+1/a
P( S > y) nP(Xl > .y) y > n ,~ .
n
(10.10) P( Zn > x) , nP( Xl > x -4
(10.11) x ^
We must now investigate the behavior of P (Zn > x) for
3/2 + 1/a 11%
1 x < n ,6^ n. It is possible to do so by the
method expounded in sections 6 and 7.
Let cp(t) be a characteristic function. A function y(t)
a.
will be called ,t-radial continuation of T(t) for the ray
t z 0, if it is defined in some neighborhood of t = 0 and
coincides with cp(t) in this neighborhood for t > 0 (a radial
continuation for t < 0 is similarly defined). For instance
KL
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
for
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
28
~; A bwt:lia ~
USE ONLY
the function q(t) g Iti (Its + 1),.,corresponding to the
11.
'Y(t) = e t(t + 1) for Vt ! Op, and y(t) =((- -t + 1) for j
1~~' 0( .
Both contittuatic is ere ttttire function's (though different ones).
Note that they 4 not coven, while cp(t) is even,
density g(x) = 2/.7r (x2 + 1)2
r\r-- From (10.1 deduce by integration by parts that qp(t)
has a radial continuation y(t), coinciding with it for t > 0,
which is differentiable at least b = ka + times. We proceed
now to calculate '(x) (the integral theorem can be obtained
later by ZQI integration). As g(x) is even, cp(t) is real/'
and we
(11.1) ?7 (x) = -= gf (,P(t) ]n --V-nn itxdt.
0
Applying the reasoning of section 5 (compare (5.9)1we getx-
n
(11.2)
P Zn
Further
ca r / roJ (11.3) (P(t) = y(t) = 1 `"
From this,
(11.k) pZ (x) _
n
Also
(11.5) 'Y(t
(P(t) ] ne- Y n itxdt + B?
(,Y(t) ]ne 1/n' itx
-c(n
dt + Be
n/fin n 1/n itx
~~ ~'Y(t)) a cat + Be
0
= yo(t) + B I t Ib;
/ b-
(q) (0) tq +
+
Y, f
q=3 q
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
fN 29
Cit. SS
e0'
For I t (s A n?/ , we have
-Y(t) = y0(
for/any t > 0,
~
(11.7) K(t) ..
-y(t) 'YO(t)
A Further, for 0 < t < AA n/ -i/n
t2' b-1
#ny0(t) _--+? -E Tq
q=3
An n/-V n, Bntb = B n-b 2
(11.9) PL
n
If we -daaeet--
(11.10) exp
where
(,11.11)
B n-b/2 + e
e
q' + Btb
+1 +e
Ad n/ -VW
exp nC
Q
Be
we obtain
~,e n
0 t zVn
B n- b/2 + 1. +e^
F 5
b [nTK,.(-,,. (t))q
K1.(t,n) = E qr
q=1.
Hence,
taking t = ~/-i/ n, _ we get
2 b-1 q
2 + qzi3 If q q!
V R itx dt
E q=3 !q tq/q
-1/2
t) = B-n
[nK3 (t)) = 1 + K1(t,, n) +
+ Kk \~ n~ ' n/ I eXp
.41 r,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
30 07E'1771~ 7
FOR
Extending the integration limit to _ oo (the error is esti-
mated trivially),
P Z (x) -x2/2 + T ~-el eXp 2
n l/2zr 0
exp(-- i~x)d~ B_n^"" ` ,-~ Tom.
The function K4(~/-J7n, n) is a polynomial with respect to
e/V n The evaluation of (11.13) is thus reduced to the
evaluation of the integrals
CO e
p
11.14) E(x,r) = Re[,- /2 ei,x rd
0
for large values of x. If r is an even number,(E(x,r
00
2) f- e2/2 lt;x rdois easily expressed : gh Armes O~
J M
e-x I'r- and Hermite's polynomials [compare (7.71)x. If r is
odd there is apparently no expression - rh?othe elementary
functions but5~for large values of x, (11.14) can be easily
evaluated by integration by parts. Thus,. for r = 1, ))o-- CA G IV
E(x,r) ti 1/x2.
We must now show that the fornula (11.13) holds for the
values of x satisfying
(11.16)
1 < x < n3/2 + 1/"/en.
Comparing nP[ Xl > x'rl n) [ compare(10.11)) to the remainder
tern B#n-b/2 +1 +e in (11.13) we deduce that n-.b/2 +1 +s
must be smaller than n(n2+1/a+s'-a _ n-2a:-Et 9. from
this,~\b/2 - 1 > 2a or b > 4a + 2. Under this condition we
obtain the local theorem valid up to n3/21+1/a +6 uniformly
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
w~, u_,_ lull
31
and hence, by integration, the integral theorem, The
relation (10.11) enables us to obtain it for the whole
x-axis.
The condition of evenness was assumed only to simplify
the final formulas; if it is not fulfilled, (11.13) will only
rG CO
involve 1 in addld tof . Moreover, the analogous limit
t, _ co 0
theorem on the whole axis can be obtained for the variables
Xi such that
(12.1) P(X. > x) f al dG(v) + 0 ( a1
. X +e,
where al > ka + 5 and G(v) is a function of bounded variation.
A
jlke similar relation must hold for negative values of x.
The new approach expounded here is applicable also to
independent variables which are not identically distributed
and to the investigation of nonnornal convergence.
IaY~~?
The asymptotic behavior of the^b*g deviations of
order statistics-can be also studied by this method.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
V Ct I,
REFERENCES
32
and
[11 S. Bochner K. Chandrasekharan, Fourier Transforms,
vviYCe bq UqderS; J
_--.--Princeton,, 1949. ~reSSJ
[] B\ . Gnedenko and A.N. Kolmogorov, Limiting Dis-
tributions for Sums of Independent Random Variables, rnbr ~~~
[A] H. Chernoff,"Large sample theory: parametric case,"
State Vol. 27 (1956), pp. 1-22.
Ann. Math.
?
" Addison- Wesley 1954.'
3
[h] H. Cramer,"Sur un nouveau theoreme M limite de la
? Act~al;tes
theoree des probabilities, AAe%oe- . Sci. 4C Ind . ,
36e!-Pvr-kk T (1938) .
[~] H. Daniels,"Saddlepoint approximations in statistics,"
y
Ann. Math. Stag Vol. 25 (1954), pp. 631-650.
of Cramer," Trans. Amer. Math. Soc. , Vol.541(1943),
PP. 361-372. ~
[9] M. Kendall, The'Advanced Theory of Statistics_,'47o1. 1,
London,n 52
[10] A . einer neuen renzwertsatz der
Wahrscheinlichkeitsrechnung;'Math. Ann., Vol. 101
(1929), pp. 745752.
l0 2.,
W A. Erdelyi, Higher Transcental Functions, Vol. 7
New York -A 1953.
[8] W. Feller,"Generalization of a probability theorem
theorem of Shannon, Us e h1 Mat. Nauk,\o1.
PP. 3-104mt (;In Rrxs an :)
Ply-
[~] R.L. Dobrushin,"A general formulation of the fundamental
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
33
[11] Yu. V. Linnik, Markov Chains in the Analytic
Arithmetic of Quaternions and Matrices, Vestnik
Letaitngrade Saver eme, Univjrsi*, No. 13 (1956 ) ,
pp . 63-68. .(-In Russ ian-.~,),
(12] U.V. Petrov,"A generalization of a 'limit theorem of
Cramer,"Uspd hhi Mat. Na Xk, Vol. 9/ (1954),
pp. 196-202. ,.(In Russian.)
[13] V. Richter, "A local limit theorem for large
SSSR?,
deviations," DoklAkad. NaukaVol. 115/
(1957), PP. 53-56. _(-In--Russ=ian.) .
[14] V. Richter "Local limit theorems for large deviations,"
I P
y
he ;~
eroydtncfi?
qrn
Vol. 2
(1957),-* 8c;:::2* pp. 214-229. (In Russian.)
[15] V. Richter "Multidimensional local limit theorems
for large deviations,"
Vol 3 (1958): XW91t-
PP. 107-114. (In Russian.)
[16] N.V. Smirnov,"On the probabilities of large deviations,"
a.Sb.,
Vol 40 (1933), pp. 443-454. -(In-
Russian.) ?txss
[17] L. Treloar, Physics of Rubber Elastics, Oxford, 0404 ohivers
1949.
[1a] J. Wolfowitz,"Information. theory for mathematicians,"
Ann. Math. Sta /\Vol. 29. (1958) , PP. 351-356.
9
[1R] M\.V. Volkenshtain and 0`,B. Ptitsin, "Statistical
F.
a
physics of linear polymer chains" Us hi gh~
Vol. k9/) (1953) , PP. - 501-568. .-(In"
Russian.:),, .
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
ru U' tl L' iwIuL Lad. UBIL I
Continuity and Conditions of Gelder for Selected Func-
tions of Stationary Gauss Processes. Yu. K. Belayev
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TO0246A011700340001-4
w ., .~a. yen L', 'r. ~CY e+r +~
Oil
HEIIPEPbBHOCTL TI YC IOB1l1H I'EJIbAEPA
AJIH BbIBOPOgHbIX MK41HiA CT400HAPHbUX 1 AYCCOBCKUIX
IIPOL1ECCOB
D. X. BEJIFIEB
B HaCTORI4tev4 pa6oTe paccMoTpeH PRA BonpocoB, cBR3aHHbIx c
JIOxaJILHbIMM CBOI CTBaMN BH60potHHx c YHHAH l cTaumoxapHbIX cToxa-
tcI114L,ecx:14.. aerrpepbIBHbIX cenapadeJILHblx rayccoBcxux ' rIpoueccoB. CJio-
Ba cToxacTwtiecHo' HerlpepblaHbi9 14 cenapa6eJILHbllri 6yAyT onycxaTbcH.
03JIO}KeHHble H14}He pe3yJlLTaTbl ObIJI14 ony6JlwxoBaxbl paHee, cm.cij,
6e3 Aoxa3aTe3ILcTB, B ? I lloxa3axo, qTO AJIR cTaquoHapHblx rayc-
COBCK14X npo eccoB 14MeeT ISCCTO aJILTepHaTWBa: 143114 HOgTIT Bee Bbd-
G0p0t HHe f yHxLNM HenpepbrBlibl, 141114 ITOgTM BCe Bb16opOgHble ~YHxuuv4
HeorpaHHveHH Ha ro60M 14HTe 3BaJIe KOHetIHOM AJI14HM. rmHOTe3a 0 Cy
IuecTBOBaH1n4 Taxok aJILTepHaTl6BbI A8BH0 BbICHa3bHBaJlacb A.H. KoJlmo-
ropoBbiM, Cm. Tame [2]. B ? 2 AocTaTo ruble ycJIOB14R AJIR HenpepblB-
HOCT14 Bbl6oporHbIx (yHKLMH, IIO]IyqeHHble XaHTOM ['3) , C~opMyJlupo-
. BaHBI B TepM14Hax xoppeJ1Ru17oHHb1x c yHHxu14il,c opMy3lbI (12) u (I3). B
3TOM }xe naparpac e IIOCTpoeHIJ 1Ip14Mepbl rayccoBcxl4x rIpoueccoB
BepOHTHOCTLIO eA14H14ua HeorpaHU4geHHb1X Ha 1IHTepBaJlax KOHeVHO 4
AROHbl. 3T14 rlp4Mepbl Iloxa3 BaoT, ZITO yCJIOB14R (12) u (13) HeJIL3R
yC11JIMTL CyIueCTBeHHHM O pa30MM. AoCTaTOgHble yCJIOBLR 4AR Toro,
Toth JOqTM Bce Bb160polqHbie (7YHxu1414 yuoBJIeTBOpnJn1 ycJloBI4lo reJIL-
Aepa paccmaTp4BaloTCR B ? 3. ABTOp Bbipa?icaeT rny6oxylo 6naroA.ap-
HOCTL A.H.KonmoropoBy za riocTaHOBxy 3a.Aav.
I AJrmTepHaT44Ba.
IlyCT ~ (t) cTauuoHapHblrs rayccoBCx14vl npol.Iece C HenpepuBHOj
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
1 ~~1 ww I NU 'iL UUL cis i
xoppe.nAgxoHHOA c yHxuxex, npxxxMamlllx BelgecTBeHHHe 3HageawH.
OCHOBHbix pe3yJIBTaTOM aTOro naparpa(a sIBJI3ieTC.9 CAeAy Iua.a
TEOPEMA I. Aim Jnodoro rayccoBcxoro cToxacTH eexHHenpepHB-
Horo CTaI]GHOHapHOro npogecca (t) cnpaBevmBa CJIeAy ulaa
a.lbTepHaTIBa: JIxdO C BepOJITHOCThm eAxHaIja BHdOpO~IHHe (~yaKgLSa
Ct) HenpepuBHH, JIxdO c BepORTHOCTbD eAuimi a om HeorpaHHgeai
Ha JImdoM xoReRHOM xHTepBaJle.
AJIR AOxa3aTeJIbCTBa TeopeMH HaM nOHaAodRTCR ABe JiexMH.
JIEMIr:A I. IIyCTI ~ (t) cxygafIHH npogecC Taxoit , nTO Ha HHTep-
sane
P{c~>>a>p..
4E4
tM ,
TorAa cylgeOTByeT xoxegxoe MHOzeOTBO Togex to I..-,#
qT0
.P{ IAACA.x ~ct~>>ca >p.
tz C.,
JIEMML 2. IIyCTb Lt) , .. ? >. Ct ~, ... , M ~St)= Q nooJteAoBa-
TeJIbHOCTb B3axMHO He3aBHCHM x CTaiwoHapHHX rayecoBCiLix IIpo-
geeooB, V EOTOpHX .i.U HexoTOporo xHTepeana - (-t', .t ) a
gaceJl >0, fi>o
> CL
>F
rAe y,,` HaXOAUTCR H3 7CJIOBHR
TorAa
(1-P)wde

kEd i - %
FOR
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR rI ONLY
joKa3aTeJILCTBO 3TNX BCfIOMOraTeJILHbIX JIeMM 3JlecdeHTapHoe x
MN ero onycxaeM.
. Kax H3BeCTHO, MOAyJIeM HeapepbIBHOCTH (YHHAZM
pe3xe A Ha3bNBaeTCR c yHxuHH
t't'~F o tit t V I c
B TOM CJlyliae, xorga Ha MHTepBaJIe & no'IT 4 Bce Bb160p0VHHe
91114 HenpepNBHb1,
940
I=0
oyHx-
AAfi Jiio6oro c, > o . B Tama odo3Hagexuxx Ha.%i AOCTaTo 1HO noxa-
3aTL, To ecru'
> pi I
; > ( )
0
AAR HeK0TOpbIX tlYICeJI t . Za >0 , r'> 0 0 1HTepBana TO
AAA JII060ro YIHTepBa.la A x Jno60r0 N(> O.
P S.%-f ('5 (4)1 > .~ =1
Em BbInOJIHeHO HepaBeHCTBO (I), TO 1CnOJIL3yfi CTa1 I4oHapHOCTL
npouecca (-) Mo}I{H0 noxa3aTb, 4T0 AJIR 4I06oro WHTepBajia
a = (4., t
1
4,0
3aMeTI4M Tellepb,
'9_2a >0 ,
rlpM HexoTOpoM
,0 =
p(t,-t.)>0
Lit (p, r) > 9 1 > P . . (I#*)
VTO ecJiw AJIR HexoToporo YIHTepB8Jla
P>o
1 P 4 Wt (n, S) > 2a. 3 > r , TO
F ..
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Y'wTILB8JI cHMMeTpHD raye0OBCxxx pacupeAenenxi oTCrAa
noJlytiaeM, RTO
p. co[~t)- ()>a
t
K3 (I') BHTeKaeT, RTO cfeKTp npogeeca c (t) Heorpawien,
TaR xax B 11 OTHBHOM cnyiqae IIOYTH Bee BHdopomie
(yHxlwld dHJii dii ge.nxH aHaxZTH,IecKmmx [4] :. IIO3TOMy
(I' HMeeT McCTO IIpx nmdoM L > o in IIpol;eeca
3Aecb (da) cnexTpajLHaa oiyzaAaas Mepa, 000TBeTCTByDo--
iAaa IIpoi ecey ' ({) , Tam m odpasoM, A.uA IIpogecca (,t)
4t()- ~JQI > o > ' (2 )
L. A
IIyCTI Ham saAaHi #>o , $ > o . Bxdepem cHanana OTpe- -
30x d=,Ct,, t')CTOJIB MaJINM, a saTeu L CTOJIb dolIiwMM, 'ITO
"~ 1'{pct)-C{,)>-~ >1-# , `" p~~ct)-~tt,)>-s~,I
tea tr0 mP(3)
rAe Yk Taxoe gene 'MCIO, 'M
Ks nevi .I H (2) IIony,4aeu, ITO AJIa HexoToporo xolequoro
MHOHeCTBa T' c
P { max [ LCt~) - tC~,)~ > 1 > . .
t, c- T
ff -, ,o b txl a L
X/ Q, ~, v-~, 03HagaeT CXOAHMOCTI B CpCAHeM xBa pawrnoM?
L
(4
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
U ` ' L L UkyiL I
n03TOMy HaAAeTcH Taxoe AOCTaTOIHO 6o.]Ibmoe /~,
S e,~x cap.)
ct Ct,) 3 > 0~ > f-
T4
Tax xax y npoitecca
C+.\ S
MI
w5, (,A , Z) > .Z0. ~ > t ,
1TO AAA
TO TaKHM ze CnocodoM HaXOAMM i& >rl, M xonexime MHOZeCTBO
T 2 C RTO AJLa
2. Z. e, 4'(t)
ir,E 1~ t
UOCTyna.a aKaJlorxgmLM odpa3oM H AaJIee, MH CTpOHM nocJIe o aTeJm
HOCTb B38.1MHO He3aBxCHMHX CTawtoHapHHx rayCCOBCXHX npoIge000a
X et
AJIA xoTOpxz
P4 C K~t~- Kct,)]>CLI>I.
Ec.im 06o3Ha iMm ~; r TIHTUB8Si(3), ~ 4),
nprdMeHs a JteMMy 2 , iierxo noitygaev , qTO
s~.~~ [C~)-Ct,~~ > a-2 > IL - ZF.
Tax Kax. E > o ? Z> old OTp630K A MOZHO BHdMpaTb CKOAb
yrOAHO EiaJIHMH, TO OTCIDAa B CBOIn ogepeAb CJIeAyeT, ITO AJM
['14111,,:' t ` 71,) ~ err ~ I
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
S4.
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
nioboro oTpe3xa
F
USE ONLY
(5)
Ha 3TOM 3axaHLII4BaeTCFi nePBb19 3Tan -Aoxa3aTeJILcTBa TeopeMbl.
i70xali{e?d Teneib, qTO h3(5) CneAyeT
P~ s L >na.\=i , H=2, 3, ... (6)
ilpoBeAeld paccyMAeHHx AAR h- = 2. Max, nycTL (5) BbinonxeHO.
CHOBa 3aAaA m cxonb yroAHo Manbie xiociia E >o , Sao . t'ls. (5)
N neMMbl I BbITexaeT, LITO cyII CTByeT Tdxoe xoHegHOU MHO) eCTBO
TolIex , "'CA, AJIf1 IOTOporo
P { MCA.)( > a
.t; C-15
Tax xax
TO MUXHO BbIGupaTm . M 0 CTOJIb 6OnLWMDJq TO 'Anti
'-~ , ~ ~ > S e +A 4 Cd'-)
~A Ie. 04
VIA CA X
tics
aAAR
S e C~ (ok)L)
i'4 21 JLCt)-J-2ct1>>
4EO
(2)
(8)
11poiAecc J(t) L1MeeT orpaHU'eHHb19 cneKTp. CneAOBaTeALHO, IIOWN
BCe. Bb16OpO Hble c YHKLkHH 3TOro npouecca HenpepblBHble. 1103TOMy
SLY
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Y lJA U 'U.ii 1. UJL U6UII. iL
OKOJIO KaaAOx- TOMKM `x F ~? '~ rte - - ? ~{'r(~ ~ 1 , MO
HO BbiAe3IY;TB CTOJIL MdJlblR OTpe3OK +s') C Q LITO
B TOM cny'Iae, KorAa 12, 11 Ct,)~ a- , TO VI BCIOAJ_ Ha OT-
pe3Ke A,,_ 12,Ct) V,(ti)>Gt Aim Bcex K OAHOBpeMeHHO C BepOHT-
HOCTLIO 60JILWeLi YMIITbIBdH (7) OTCIOAa noJlylLaeM, TO
.n~
h { V [ ~,~ ~, Ct,.) -- ,ct,)) > a ] } > 1 - Z (9)
k=2. {ELK.
PaccMOTpll;A Tellepb npouecc ~2Ct) . ioBTOpwB paccylicAeHMH RepBOtt
gaCTYI TeopeMbi, MO?KHO noKa3aTL, WO AJIH OTpe3KOB A
1/13 3Toro paBeHCTBa m Via (8), AJIH COGUTIdH k,. COCTOHiI(ero B
TOM, LITO Sr C -~ZLt) - 7=Ci-. > cl- ~~ .
t(416
UIMeeM
co~.
1zc.~,~-s~) - %2a(t,~ -s 1 -
RyCTL SK C06bITMe, COCTOHI?gee B TONi9 PTO AJIH Z. t < k
L 4 [ ~~ C+) R, , r` k [ ~,C~) - -~, (f,)] >0
B 3THX o6o3HageHw x (9) nepeiUUlcblBaeTCH B BHAe
Ytn TbIBaH (9' ), mmeeM
P{i Z O L
4,
P { V, 8 ,AV*
(9)
?
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
77r.0,7' r,
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
A U11 J~ij~ ' UVL UIIL I
_ 2P BAk1- Z- P ~ AK >., s - F.
?~-I
R_1
Tar, Kax E ?O 1H S > O M0 KM0 Bb1614paTB CKOJIb yroAHO MaAhIJ1M,
TO OTCIOAa cAeAyeT BbInoJIHeii a (6) AAR n=2. IoBTOpWB Te ae pac-
cyicAeHwR, MO KHO 6bvlo 6u, ucxoAR 1i3 (6) npli 'h-=.2 Aolta3aTb(6)
AAR11.=4 vl T.A. CAe4ljOBaTeJILHO, AAR Aio6oro oTpe3Ka z AI06oro ./t> O
2 k c-~) >
TeopeMa 4oxa3aHa.
? 2. lOCTaTOgHbie yCJIOBIdR AAR HenpepblBHOCTM.
11pv1Mepbl BCIOAy HeorparimeHHux npoueccoB.
PacCLIoTp11M Tenepb AOCTaTOlIHUe yc3I0BoR AAR HenpepblBHOCTM
Bb16opO1xbIx (yHKU14 I cTaujoHapxblx rayccoBcKKx npoueCCOB. Hav16o-
nee C1IALHbI9 pe3yJLTaT, 113 M3BeCTHbIX B HaCTORluee BpeidH aBTOpy,
npI4HaAAeM1iT XaHTy L33 . ;QocTaTOUHbie ycJIOB1R, nonygexHble. XaH-
TON, c(opMynwpoBaxbI B TepMI4Hax CneITpaJIBHb1X (yHKIkMk. AJIR
nepe(opMyJIipOBKM 3TVIX yCAOBM9 B TepMI4HaX KoppeaHgmOHHbIX (yHK-
u1Ie1 oxa3b1BaeTCR nOAe3HOY c ieAyiouaB
JIE''. MA 3. EcJni 'F(C-) Hey6blBaloiuaR (JyxxuHR orpaH1qeHHO i
BapMauvlll, a .
~eC~l =SCi-
TO 93 TOrO, qTO AAR HeKOTOpbIX `>.O
TogHO MaJlblx &
ago
cAeAyeT, 'iTO np1 A106oU Qo < a.
< o0
U LJLLtLA= UWL
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 06paTHO, eciw (II) BbinoJHexo RPM HeKOTOpOMM 6>0 , TO (10) capa-
BeAJIMBO npM mocbix a , e > o , AAR Bcex AocTaTO'iHO MaJlbiX
XaHT noxa3an, cTo eCJIYi cneKTpaJi Hax cpyHKL 1fi FCa.) cTai o-
HapHoro rayccoBcxoro npouecca (t ) Y4OBneTBOpReT ycnoBMIO(II)
RPM HeKOTOpOM 49 > j , TO fOgTL1 Bee Bb16opogHbie c yHKUMM npouecca
(t) HenpepblBHbi. C Apyroii CTOPOHbI, B my . ,yxa3aHHOi: BM1lie JleM-
mu 4 M3 (10) npM 0. > 1 , cJieAyeT BbinOAHeHKe (II) Aim
< 4 < ct . 1I03TOMy yCJIOBMe (10) RPM 0. > 1 Tame AocTaTOi-
Hoe AnR HenpepbiBHOCTM BbIdOPO IHbIX c yHKutdii. Ta.KI o6paaoM, cnpa-
BeAnMBa cieAyioiuax
TEOPE MA 2 AJIR Toro, MT06b1 fOqTM BC O Bb16opouHbie c yHKLHH
cTaijmOHapxoro rayccoBcxoro npouecca ~(+) 6biJIM HeripepbuBHbI, Ao-
CTai'O4HO, 4T06b! 6b1JO BbinOJIHeHO OAHO M3 cJieAyiou x 3KBMBaneHTHbIX
IaeicAy co6o9 ycnoBMg:
MJM apW HeKOTOpOMa P > 1
00
F C.X) < oa (I2 ).
o
HALM npL HeKOTOpbD( 0.> I , C > 0 AJIH BCeX AO.CTaTO iHO MaIbIX Ft~
MI Ct+~.) ~t)r s a.
1 III
BbIBeAeM TenepL HexoTOpb:e AocraTOgHbie yCJIOBMH AJIR Toro, tITO-
6bi fO TM Bce BbICOPOtIHble (, yHKLMM CTaIMOxapHbix rayccoBcKmx npo-
ueCCOB 6bIJM HeorpaHMtIeHH Ha JI106oM xoHeVHOM MHTepBane.
TEOPEMA 3 Ecri y cTaLuioHapHoro rayccoBcxoro npouecca fi(t)
cyII ecTByeT cueKTpaJLHaH IJOTHOCTL p.) T3xafi, qTO AnR I3eKOT0-
ILY
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR DFeUiAL SLY
pTIX 0 a0 0 M B c e x
1 > ~o
TO IIOtITM' BCe BbI6opOtlHble t yHIMMM HeorpaHI4 eHhI Ha JII06OM MHTep-
Bane KOHetmo 1 AJIMHbI.
McTO4 AOKa3aTeJILCTBa 3TOLl TeopePdbt COCTOVIT B CJI6Ay1OU M. 11yCTb
B3aI4MHO He3aBMCWIdble CTaLIMOH pxble rayCCOBCRHe npoueccaI
M ~ (t) = n = i, z , ... TamMe, LITO Y HMX cyAeCTBy1OT
CneKTpaJILHble IIJIOTHOCTM l n (;') AJIR 0 - Zh
to
Pacc;1oTp ii Tenepb cJlytlai HbIt npouecc >ZCt) _ ,~ CM
04 bo
C" C AJIR Bcex > IN M Hexo-
TOpOro a I >0 . VIOXHO IIpOBepMTL, LITO AJIR 3TOrO npouecca
P f C', I ; . Z-
0
N3 AOKa3aHH0LI B ? I aJILTepHaTMBbI BbiTeKaeT. ITO rO'ITL Bce
BbI Opo'Hble ()yHILMH npouecca I C+) HeorpaHUILIeHbI Ha JII06oM KOHeti-
HOM MHTepBane. 1!.JIR CneKTpaJILHO IJIOTHOCTM q (x) cnyt;a4Roro
npouecca C Ct) AJIR HeKOTOpbIX 0 C k, c k2 MtaelOT M CTO Hep eHCTBa
TaiM:a b{e CB09CTBOM HeorpaliWtleHHOCTM o6JIaAaeT M CTaWMOHapHb1t
rayccoBcKMLi npouecc jw (t) y KoToporo cneKTpaJILHaH IIJIOTHOCTh
I,WC-X)z 0 AJIR X '- H ~/ C)) _ OL) AAp A > w' . 3araegaH
Tenepb, Llro MCXOAHbIL! npouecc ~ (t) MO ECHO npe;1(cTaBMTb B BMAe
CyMMU AByx B3auMHO He3aBMCMMbIX CTawIOHapHbIX rayccOBCKHx Ilpo-
ueccOB, OAMH M3 KOTOpbIX MMeeT cneKTpaJI HyiO IJIOTHOCTL BMAa d? Jw(4
0C > 0 , vJ>O, noJlytlaeM yTBeptAeHoe TeopeMu.
V
fin
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
11 -1 w:! ue I IWIL UQL ui L r
IIpvlMepbI HeorpaHHgeHHux rayccoBcK'x npoueccoB MOXHO Tame
CTpOLTL, MCXO H M3 CBOir CTB KoppeJIxu1oHHbIx c yHKW I.. BecLMa noAe3
Hove oKa3bmaeTCR cJieAycoutaR Jie ma, Aoxa3aHHa1 A.A.BeHTueneM.
JIEiAMA 4. IIyCTL , ... , .,, cJrygaMHb1e rayCCOBCKI4e BeJ1M Hma
M 0 M~.t. 6z M 4 0
. a _ xt
V% i 7==Y= V -Oo
ECnm xoppenxunoHHa c yHKUWi cTauuoHapHoro cJlyqavlHOro' npo-
uecca (~) 13 ('~) - {t ) L4~ BbInyxnaR uJrR 0 5 ~ s l',
>0 .TO
0, I+2-tII 'F.
lOJIb3yICL JIemmoz- 4 M02KHO noKa3aTI, XITO eCJI1 y CTaI ZOHapHOrO
rayCCOBCKOro npouecca (f )
M I ( + C1,~ - (.t Z ' C > 0, (14)
AAR Bcex AOCTaTOgHO MaANX ~.i
a $( ~.) BbInyKJIaR c yxxuvla, To
llp meHRR Aanee aJILTepHaTMBy ? I nonyiaeM, ITO Bepxa
TEOPE: A 4 Ecnyi y cTagmoHapHoro rayccoBCKOrO npouecca ~(t )
KoppenRuuoHHan c yxxuMR 6( .) Bbryxna 1 AJIR $0>0 i
LMeeT M CTO (14), TO fO 1T4 Bce BbI opoqHbie c yHKL 114 npouecca
(t) HeorpaxvNeHbi Ha niodoM v1HTepBaJie KoHetIHOR z JIMH I.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
? 3. YCJIOB1R renbAepa
B cJlygae HenpePb1BHbIX C BepOHTHOCTLIO eANHUua CTaLHoxapxblX
rayccoBCx1x npoueccoB BCTaeT 3aAaga 13ytieHMR MOAIJIR HenpepblB-
HOCTUI BbI60pOtIHbIX (yHKUUtV1. 1/I3 o(u.,1X pe3yJILTaTOB MO)KHO OTMeTYITb
cileAyiou>, 4i . EcJIU K)> 111)1 y 0 , TO BepORT-
HOCTL CO6bIT1H l}paBHa JIM60 . HyJI1O, AH60
e,qUIHNI.e. IipI,IMbixaeT K 3TOMy Me xpyry BOIPOCOB u cnaAylouiaR Teo-
peMa, HBJIHIOUtaRcH 060611I,eHM M pe3yJiLTaTa BaxcTepa [5] , Ha cJly-
gag cTagzOHapHbix rayccoBcKmx npoueccoB.
TEOPEMA 5 HyCTb 1,2 ABa CTaiwoHapHbie rayc-
?
coBcxlle npouecca M TaKL4e,
z
~ITO 4- ~ -, 0 , a of > j . TorAa HaliAeT-
`4` O o)
CH TaxaH noc3Ie4oBaTemLHOCTb 'IVICeJi ?y1) , yt .-, ov , iq TO c
BepoHTHOCTbIO eAHHNua
,4oxa3aTeJILCTBO 3TOl"4 TeopeMbl Id0)KHO 110JIyV1Tb, 1Cn0JIb3yfi
TOT 0axT, PTO AA R CTaw oHapHor.) raycco c oro npoueeca ( t
He AmiiepeHLupyeMoro B cpeAHeM IBaApaTI1 HOM 3HageHHa
ticol e"
n%-# 00 VA = n_, I 1, a. -O. Wt L d
LkA Q - t7 ~-k) .
M 1 ~~tk,)- ct)
--~ (Jrikr~) - ~ (+)
FM Ct~ ~z) (})f
CTaHOBRTCH aICUIMnTODTYIveCKM B3a1MHO He3aBHCMMUMM CJIy~Ia4HbuMH
BeJI14g1HaMM, KorAa 1 , _ (.sw.f , a hs ~0
..CxaieM, VTO cnymaLHbllk npouecc (t) E- (cc~C), ?o cocc j ,
C > o , ecJIM AJIR JI1o6oro C 1> C c BePOHTHOCTLIO eAUHHIa paBHO-
.MePHO 11o BCeM t z3 xaaAoro WHTe1BaJla A KOHe4HOYI 1IIMHbI,
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
MR 0 L
cyuteCTByeT Taxoe F (Q C), qTO AAA Bcex '~ c V (Q C)
I ~' (+t &) - ~ ( ) I c' I W.
Cne,gymuAaa Teopersa capaBe4JIwBa AAR npoueccoB o6utero THna.
TEOPE ti1A 6 AAR Toro, qTO6bi cTOxaCTwgecxo HenpepbIBHbIM cnygag-
Hu 1 npouecc (f ) E. f4(4, C) AocTaTOIxo, gTO6bI
uo
IPfl() - ~-?~~ l > K,z C. Z-,a } < oc, (I5)
h=1 X=a
rAe EcJrv rIpM HeKOTOPOM 0a > 0
M I BA) - c (i) I < l(&) ~- o Hp M h -s 0, TO M3
oO
YtaCL ~~-;,. L< 040
h.~
CJle,gyeT, tITO F H (o, C) AAR nio6oro C
>10.
AoKa3aTeJLCTBO 3TO i TeopeMbi BROJIHe aHaJIori qHO Aoxa3aTeALCT-
By N3BeCTHoYi Teopembi A.H.KonMoropoBa 0 HenpepbiBHOCTUI Bb16opoq
HbIX c yHK I1 [61 , CM Tame pyccxHI nepeBOA KHmri4 Ay6a [?
CTp. 576. -
TEOPE.V1A 7 Ecnu xoppeJRuvioHHaI c YHKW R 8(1i,) cTawwOaapaoro
rayccoBcxoro npouecca TaxoBa, 'qTO Aii BCex AOCTaTO4HO manbix
M I (44&) - ct~IZ
J.P'j 12 Of.
e' I.e,,tIII ' (I6)
TO Ct) E H (S > 13 (~.
~Cd / >
Bbinyxna1 (YH1UUIH Tama, WO AAH BCeX AOCTaTOVHO MWI MX N
0< CL C1
C2- C I '~. I (I? )
e1~
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
TO 2C , HO 0 H BCex AocTaTO'IHO MaJlblx -' s S (ee,
U1MLY
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Tq -
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
J b 1i lei IIIJ1AL UAL SLY
uideeT McCTO (I6), M yTBepri{AeHNe TeopeM u 8 ABJIReTCR CJieACTB1eM
TeopeMbi ?.
AHaJIorLIgHbIP.I nyTeM Aoxa3bIBaeTCx CJIeAyiogm pe3yJILTaT, Bnep_
Bble ' noJlygeHHbl9 ApyrMM crlocodOM XaHTOM [3 J .
TEOPE'MA 9 EcJIM cnexTpanbHaR c)yxxu4H FCt) y CTauuoHapHOro
rayccoBcxoro npouecca (t) TaKoBa, qTo
to
S ~2a~ NCB) oa ,
0
TO nogTM Bce Bb16opot1Hbie (yHKLMM y,4OBJIeTBOpRJoT 0606UHHOMy ycno-
BMb realAepa BMAa
1
T Le, I ) 7-
paBHoreepxo no Bceid M3 Jfl0 OrO MHTepBaJla KOHeRHO2 AJIMHbI,
4JIR n-o6oro C > o M Bcex AocTaTOMHO MaJIb1X
B cnyqae cTaL oxapHblx rayccoBcxnx IIpoueccoB, HenpepblBxas3
Au( (epeHL[wpyeMOCTL C BepORTHOCTbio eAMHMua 3KBMBa]IeHTHa1 yCJi0B1410
iIwnwin. a. ,jeYicTBMTeJIbxo, eciM AJIR 990CTaTOgHO Manbic paBHoMep-
HO no M3 HeKOToporo MHTepBaJla KOHetlHOR AJIHHu C BepOHTHOCTLIO
eAMHMua
1 ct+~.). - (i 1 < CW I I,
me %>0 cnygagHaj Be)IMtnHa, TO noqrH Bce Bbl6opogHbJe (yHx-
L>;MM a6coJnoTHO HenpepbIBHH. HpMqeM npOI3BOAHaH, KOTopaH Tame
HBJIHeTCH cTa4MOHapHbIM rayccoBclcw m npoueccoM orpaHMgexa c Be-
PORTHOCTLIO e,LMHMua. HeilpepbIBHOCTL npoM3BoAHoI RBJIRerCR cJIeA-
CTBMeM TeopeMbt I (? I)?
B 3axJIloLIeHMe OTMeTWM, VTO y rayccoBCKMx npoueceoB, xoppe-
JIRL1OHHbie (yxxL[MH KOTOpbDC aiiaJIMTM'IecxLe, IIOtITM Bce Bb16opo4Hbie
P, PT, -I ; rr
r. Y
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
iirt urr biAL uaC uRLI-
c yHKAmm aHaAZTXgeCKZC, 06JIaAaloul a pHAOM CBOMCTB CM. (I,
[4]?
A, 1I T E P. A T Y P A
[I] 10.K.EemlfieB, JIOxaJIbHb!e CBO' MCTBa Bb16opo4Hbnc OyHRARR eTa-
40OHapHblx rayccoBCxMx npoueccoB, Teop.Bep. vi ee rlpwMeH.
T. 5, BbIf.I (I960)
[2] P.JI.AO6pyM14H, CBOhCTBa HenpepblBHOCT' BM6OpOtIHb1X (yHXL L L
cTaLi oHapHbjC rayccoBCK14x npoueccoB, Teop.Bep. H ee npH-
meH. T.5, Bbrn.I (1960).
C3 G.A.~uk'~' , Kctrio(owA Fo j e'. 64VIS fol-KUS)
TRaws. Aw:A., 144o~&. Soc. J. I (1951 , 38- 69,
.(ECTL P CCKN nepeBOA c6.MaTeMaTUxa 14 PIA, T.2 i~ 6(1958),
87-114).
[4] i0. K. BeJIReB, AHaJILITI'PIecKMe ' cJlytza9HLIe npoueccbl, Teop.. Bep .
H ee np4MeH., T.4, Bbin.4 (I959), 437-444.
[5] Cr. tuXtC , A S~zoK '~ ~u.; ~' t~eo vK fez.
av. %ocR4V-S , 132oc A wt h . Yv1 of Jt , Svc . v/.
n. 3, (I956), 522-527.
161 e .t. Se,,(,I , Ae .), 1>0 Ct-0 sit. L t",4 O(?
fv-A,%VOoz of A.~o'ti~ , Crto'tn. A i -- : g(17n)
1f3t1
(ECTL pyccKj nepeBOA, TpyAb1 Cp.A3.yH-Ta, cep.MaTeM. (5),
3I,'(I939), 3-15.
A.Ay6, BepoRTHOCTHue npoueccM, O4'IJI, 1956.
MaTeliIaTINecl+:. 14HCTMTyT 4M.B.A.CTeKJIOBa, r. docKBa.
.~1'n IR O.b. r'~'':: '4 fl & 17 W9P`. ^" h \'I 4d
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
by B V Gnedenko
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
'IV
ILI
fu, wildbu?
Anexcaii p YixoBneBHc X HRNII
AaMbI H rocnoAa!
CeroAnn mu co6panLcb 3Aecb,RTO6bI OTAaTb Aonr IIaWRTH MaTe-
MaTHKa, 3acnyrYi KOToporO B (OpH1'!pOBaHHH OCHOBHbIX HAeL H McTOAOB
COBpeMeHHOI TeOpLH BepORTHOCTei IIpL3Ha1TCR BceMH.
OAHOK H3 xapalTepRux oco eHHOCTeUI pa3BMTHFi COBpeMeHHOR
HaygHOili a icnH RBnf1eTCR 6ypnutl pOCT CTaTMCTHgeCKMX KOHI;WIII;HIt
B pa3nItuibnc 06naCTSOC eCTeCTB03HaHHA,TeXHHKM M 3KOHOMMXH.C nOJI-
RO4 OupeAeneHHOCTbIO BbIHCHHnOCb B,ZITO npHBnegeHHe McTOAOB
T80pHH Bepo$ITHOCTe Ii K H3ytIeHHIO npHHIj4nLanbHb1x upo6neM cm3HKN,
6HonorHH,xkMMH,aCTpOHOMHM,a TaICKe 3KOHOMMKH RBf1eTCR H8 IIpH-
XOTBIO OTAenbHbDC 1ccneAOBaTenel i'l He npexow e11i MOAota,a HeH3-
6e?KHOCTbI0,BbI3BaHHoM cynJeCTBOM Aena.B pe3ynbTaTe' .Tenepb o6ocHO-
BRHO C4MTa1OT,XITO 3acOHbu IIpipoAb1 HOCSIT CTaTHCTMtiecKMYi xapaxTep,
o6ycnOBneHHFM AMCKpeTHbl.M CTpoeHMeM MaTepHM.M3BeCTHO,t1TO 3Ta
TOt1Ka 3peHHR nocny?xina OCHOBOt MHOrOt1HCJ:eHHHX yciexoB BO BceX
o6naCTRX Hay1M.CaMO co6oR pa3yueeTCSI,t1T0 yxa3aHHoe 06CTO$ITenbeAP.
Y
CTBO AOJIZHO 6bMO CKa3aTbCR Ha H3MeReHMH COAepZaHHR TOR BCTBH
MaTeMaTHKH,KOTOpaH HMeCT CBOeK qenb1O H3ytleHMe cnygaYiHbiX sBne-
u .TeOpHR BepOHTHOCTeM He Morna OCTaBaTbCF1 B TOM COCTOAHHH,
B KaKOM OHS HaxoAHnaCb B npOMJIOM Bexe H gaze B uepBbie ABa Ae--
CRTMneTHR HamerO BeKa.ROny1HTy13THBHMY fOAXOA B WMW onpeAeneHMH
OCHOBHbIX UGHRT14 TeOp1H BepO1THOCTeM,xapa1CTepHI4k AJIR 18 x.19
CToneTwt,He or yAOBJIeTBOpLTb He TOnBKO MaTeMaTHKOB,HO H npeA-
CTaBHTe3Iei4 eCTeCTBO3HaHMR.& &73onLpOBaHHoe nOno' 6HHe CpeAH Ma-
TeMa.TM%1eCKHX Hayic,KOTopoe 3alruana TeopLR BepOATHOCTeM ei a co-
BCeM HeABBHO,HaXOAHnoC1 B pe3KOM npOTHBOpet1HH C T014 OTBeTCTBeH-
Hid Befieget;jISM, KaKj cTana OHS BO Bceu cICTeMG Hajlx
HMA
Be
3H8HHj.B pe3yfbTaTe OAHOkj is C MLi c nepnoogepeAHb1X eH-
For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
offly
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 2 -
MIX 3aAag, Cn0? HBIIIHXCR RO BTOpOMyye CRTHn8 THN HamerO Bexa, oxa.+
3anaCb 3aAPt.a npeo6paXOBaHHR TeOPHH BepoRTHOCTett B cTpo iHYM
MaTeMaTHgecxyl0 ANCUYInnmHy c norxgecxH OTTOgeHHbIMK OCHOBHW M
nOH$ THAMN, C MHpOKO pa3BHTMMH CneiXm Hgecx i McTOAaMH
HCCneAOBaHHR,X Re TKO yCTaHOBneHHM6aH WHIRMIMMEM CBR3RMH C ApyrH-
?&M B8TBRMYi MaTeLaaTH}H.gTO6bI TeopHR BepORTHOCTe' npeBpaTHnaCb
B Aer4CTBHTCnBH& McTOA Haytworo nO3HaHHR EPHPOAM H806XOAHMO
6BIfO Mae IIIYlpOxO pa3BHTb ee npo6neMaTHKy,rny6oxo npOaHanH3H-
pOBaB OCO6eHHOCT14 nOCTBHOBOI{ MaTBMaTHtiecxHX 3aAae eCT8CTBO-
3HaHHR.
Ponb A.Fi.XHHtII Ha B peuieHHH Bcero KOMnnexca TOfBIO gro yxa-
3aHHBIX BOnpOCOB HCRnmtiHTenbHO Benuxa.K HBCMOTp$! Ha pa3HOo6pa-
ero
3He l~iaytiHMMX HHTepeCOB,O8H npOH3BOART BrlenaTneHHe eAHHCTBa a
HaytIHOYI geneycTpeMneHHOCTH.B 06ill4X BepTax xx MO iaio oxapaxTepH-
3OBSTb xaK ID +mmzzmm p
I= CHCTeMaTillgecxoe 143ygeHHe MECTa H 3HaReHHR CTaTHCTHgecxxx
3aROHOMepHocTe14 B pa3nxgHblx tiaCTfDC MaTBMaTI4j1,# ecTecTBo3Ha-
Hi H TeXHHKH.A HaAeI0cb,tiTo ,TAanbHei!uee fOATBepAHT CKa3aHHoe.
AnexcaHApcoBneBHq XHHg1H poAHnCR 19 Hmnm 1894 r. B Co-
ne KOH pOBO MMeAbmcxoro ye3Aa KanyxcxoK ry6epHHH,H3BecTHOM if
B Te BpeMeaa cBoett 6yMaZHOV"c c a6p1xotI.Ero oTeu rio Cnet;HancHOCTH
vm eHep-Texxonor 6bin rnaaHb>isa HHneHepoM Ha yxa3aHHOK ~a6pi ce w
cpeAx cnegzannCTOB B o6naCTUI 6yMaroAenaTenbHoro IrpO13BOACTBa
nOnb3OBancs I43BeCTHOCTBIO 4 aBTOpHTeTOM.,QeTCTBO,a 3aT8M id X8-0
HHxynRpHbie McCRLj& B riepHOA 06ygeHHR CHanana B peanbxoM ytiLn1
uce /MOCKBa/,a 3aTeM B MOCKOBCIOM yHHBepCHTBTO A.$i lpOBO Zf
B KOHApOBe.TaM OH opraHH30Ban nIO6HT8nbcxx4 TeaTp,
ygaCTHHxaMH KOToporo CTanl ero CBepCTHHKH143 Cp8Abi pa60THX,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4 77,
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
3T14 roA& paHHe 4 IOHOCTM 6bdnN O HOBpeMeHHO roAaMN yBneneuR
nNTepaTyPO N CO6CTBeHHBDC fO3TNgeCKNX npo6.Pe3ynbTaTOM 3TMX
yBnegefl4R FLBNn1Cb' HeCKOnbXO TOMNKOB CTNXOB,I43AaHHBIX B nepNO,A
c 1912 no 1917 roA.HecoMHeHHO,t1TO yBneneHNe TeaTpOM M nNTepa-
TypOA oxa3ano orpommoe BnLlRHNe Ha ()OpMNposarne XNH n Ha man
OAHorO M3 CaMLIX 6neCT$MMX neITOpOB,BeAarorOB N aBTOPOB MaTBMa-
TNgeCxOY nnTepaTypbi.Kai B yCTHOM,TaK N B n1CbMeHHOM H3noi$eHNM
OH yen MaCTepCKN CoceTaTB npeBOCXOAHyIO nITepaTyPHy1 (OpMy C
HaygHOYi rny6NHoII, OTgeTn1BOCTBIO N $TCHOCTBIO MBIC IM*
C 1911 no 1916 r. XNHtIMH 6bUI CTyAeHTOM (l13Nxo- MaTeraaWtlecKf
xoro (axynbTeTa IMOCKOBCKOrO yHUlBepc1TeTa.TaM off npYIMKHyn x
To a rpynne CTyAeHTOB,xoTopaR 6bma ysnevexa NAeSIMII TeOp1M t yHx-
Ij1ld AeYICTBLITenbxoro nepeMeHHoro m pa6oTana nOA pyKOBOACTBOM
npo~eccopos A.4.Eroposa N H.H.7Iy3NHa.IIepBble ero caMacTORTenb-
Hole HaytlHble nand >fie OTHOCRTCR Kax pa3 K 3TOMy BpeMeHH M 6bl-
nN Bbl3BaHM N3BeCTHMIMI4 pa6OTaMl4 A.AaHxya 0 np1MNTMBHEIX c yHxLH)iX.
B AoxnaAe,npogITaHHoM 6 HUR6pR 1914 roAa Ha CTyAeHtIecxoM MaTe-
MaTHRecxoM xpyxKe,XNHtII4H npeAno1Nn eCT8CTBBHHOeA o6o6igeHMe ?IIo-
HRTI4R npo13BOAHOM. finR Bcero AyXa NAevi T80p:'Yl (~?yHKl;I4Yt~ 3TO nOHR-
TNe IlpotlHO BOMnO B apcexan cOBpeMeHHOYi HayKN noA HaIMeHOBaHLIeM
aCgIMTOT14 CcxoM np0I43BOAHOL .IMffi LMH>l~II IIOA 3THM Hal4MeHOBa-
HNeM OH Ilpe,nO?ildn nOHNMaTB CneAyloi1ee:eCnl4 B TOtlxe x, CyIg8CT-
ByeT npeAen
Q,,?,,, f Cx) - ~(x?)
x~x. x - ar.
(L)
xorAa x np1 CTpeMneHNN x %a npo6eraeT 3HatieHNR,IIPLIHaAJIexaIl(Ne
HeKOTOpOMM MHOxeCTBy E , NMBlolgeMy B TO'IK8 xo IInOTHOCTb 1, TO
3TOT npeAen Ha3BIBaeTca aCNMnTOT1geCxON npo13BoAH01 7yHXIjl4$
f(%) B TOWS X. B yxa3aHHOM AOKnaAe OH nOKa3an,gTO Tax onpe-
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-4-
AeneHHOe IIOHfiTHe NHBapmaHTHo OTHOCNTenbHO Bbi6opa MHOIecTHa E.
HHb1GdN CnOBaMN,ecnh Kax a no Aga MHo ecTBa E1 H E2 IMeIOT B
TOtn{e X, nnOTHOCTb it npeAen (1) CyMeCTByeT Kax A E,1
Tax N AI$ E2,TO O6a OHN paBHbi Me IJ CO60I.HOHRTH8 aCZMIITOTMgec-
K01 IIPOH3BOAHOIA N ee NCIIOnb3OBaHNe AfH geneR O6O6igeHNR nOHRTHJI
NHTerpana ile6era 6brno npegMeTOM IIepBr1x Haylimix CTaTeI2 XNHtuma.
flo3AHee OCHOBHaH NAen 3TOrO nOHRTNR 6bina mMpoxo NM icno m3OBaiia
AJIH BCCCTOpOHH8rO N3yg8HNH JIOxaJIbHoro nOBeAeHNH N3MepHMbiX (kWHIC
uN~?
rOBOpRT,gTO HBKOTOpOB CBO CTBO OCyigeCTBnHeTCR.B.AaHHO TOtI-
Ke aCM,'JnTOTNgecKH,ecnH OHO NMeeT MSCTO IIOCne yAaneHNR MHO?IteCT-
Ba,HMemigerO B HeII IInOTHOCTB O.XNHgHH npeAnoh'{Hn Ha3bHBaTb iyHK-
LjNIO -(.a) acHMIITOTNgeCKN HanpaBneHHOYi. B TOgxe x , eCnH OHa CTa-
HOBNTCH aCNMIITOTNgeCKM y6b1BaIoige2,Ba3pacTaIoigeta N3IN IIOCTORHHOA.
(tyHKIANH T(x) aCNMIITOTNge CKN nanpaBnena Ha- AaHHOM MHOXe CTBB - IIO
nO3NTenbHOYi Mepbi,ecnx OHa acH nTOTNgecxm HanpaBneHa nORTN BO mmIE
Bcex erO TOMMX.OCHOBHOA p83yJMTaT,BbMCHHMWA9 cppoeHlda acZMIITO-
TNgecxH HanpaBneHHb c (7yHKI;NIri AaeTCR cneAyloigel TeopeMot:tITO65t
(yHKi a P(x) 6bina aCNMIITOTNgecxz HaripaBneHa Ha AaHHOM MHOXeCT-
Be,HeO6XoANMo N AOCTaTOgHO 1TO6bi ee 3HaxieHNR B 3TOM MHOKeCTBB
C TOtmOCTbIO AO MHOxeCTBa IIpON3BOMHO Manorl MepH COBnaAanz Co
3HageHNRMN HenpepbrBHOYt ~yHKIj1N,o6naAaioigeA n1mb KOHegHW gmcnOM
Ma(CNMyMOB N MYIHNMyMOB.
Ba? iOCTB IIOHRTIIIR aCNMHTOTNgeCKoi HanpaBnerHOCTN noAcepxNBa-
eTCH TeM,tTO IyHKIWH,06naAMomme 3TNM CBOLCTBOM,HMeIOT IIOgTN BCIO--
Ay Ha paccMaTplBae:li0M MHo, eCTBe acHMIITOTHnecxym npON3BOAHy1.Yc-
JIOBN8 CylgeCTBOBaHNR acMMIITOTNgecxo IIpOLi3BO=0IIt IIORTH BCIOAy Ha
OTpe3K8 6MnO Ha.II~AeHO XNHtlHbM 8U 8 B 3aMeTxe 1917 r R 3TOro
1917,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
R L 3` U~
-5-
H806XOAKMO H ]. OCTaTjaqHO, LIT06bI ,'[(axxasi (_jmci;msi coBnaAana c He-
npepblBHoi (yHxIHert orpanFIqexot BapI4arYIH Ha Bcer OTp83Ke 3a Hc-
KntogeHHeM 6BITb MOiCeT MHOZeCTBa CKOnb yroAHo Manod Mepbi.061gax
cTpyKTypa m3Mepmmmx ciyHKgHH BM BnsieTCA cneAyiolgiM npeAnOiceHHeM
XYIH.iz4Ha:BCRxag H3MepL1Man (JyHxLH a HCKJIIOg?HHeM,6biTb 1oEeT,MHO-
Ze CTBa Mepw Hynb, nM6o HMe eT aCLlMIITOTI4ge CKyI0 IIpO13B0AHyi0, 3HIdO
o6a ee BepxHJ x a C1MIITOTLIgeCKHX IIpON3B0AHMX gI4cna paBHM + eo ,,
a o6a HHZHHX a CLlMIITOTMReCKLIX IIpOL13BOAHb1}C gHcna paBHw - 00
Bcxope nocne ony6nmKoB HMR pa6oTw I'MccneAoBaH1RR 0 CTpowimn H3-
McPM1,MX (yHXI;Mg" B 7icypHane MaTeMaTligecHmii C60pH1K,B KOTOpOi.
fl OBOAH ncH yxa3aHHM4 aHanvl3,ee IIOnHbrili nepeBOA Ha (paHL 3cxM1k
R3fl 6wn ony6nMHOBaH peAaxg1e t Xypxana "pundamenta Hathematica"
173Mep1MbDC
3T0 yBnegeHHe HCCneAOBaHHeM rny60HHX CBOYICTB IQl$IIIPIII~JHKWHII
He TIpOmnO 6ecc ieAHO H14 AJIR MaTCMaTLlK14,H14 AM Bb16opa nocneAylo-
ti x HailpaBneHHki pa6oTw Camom XVJH1WxwM:nP0A0zzeHHe pa3BHTHR
LIAetR X1Htidxa,oc06eHRo AnR cny'iaa (ljHKgHM MHOrMX nepeM2HH&x,
OCyTgeCTBJIReTCR pHAOM ygeHb1X H B Hammes AHm; 6 1
AaJIBHeItm1e pa6oTw caMoro X1 H 1Ha xax B o6naCTld Teop& i tmcen,
Tax H B TeOpMH BepORTHOCTeYi B 3HagI4TenmHOA CTeneHH IIpOBOAH-
f1Cb IIOA BTL4.IH14eTA ero nepBOHazIanbHMX I4HTepeCOB.
~M f OPIMPOBaHHR XMHc1Ha xax ygeHoro HCxnlog1TenbHO 6oJib-
moe 3HageHme misaJm 1929- 1925 r.r.,xorAa off Hagan pa3pa6oTxy
AByx HanpasneHM9 MaTeMaTLfgecxIx 1ccneAoBaHHVC mmpoxoro MaTeMa-
THgecxoro 3HageHY1R.06a 3T1 HanpaBJleHHR MaTeMaTIgecxoA Mwcnr
B 3agaT0nH0M ~3OpMe iniejiI4cb y3ce y 3. Bope7IR.OAHO 143 H14X &MO
CBR3aHO C CHCTeMaTHg3Cxl4,d Y13ytI8H14eM McTpIt eCKL1X CBOYICTB pa3-
jimmiL1X KnaCCOB ldppagloHanbHLDc gmcen,gpyroe - C C14eTeMaTHtec-
x14M HCIi0nb3onaHHeM CpeACTB H II0HRTI4L Teop1N (ymcL Mt l1 Te0p114
MHO-SeCTB B TeOPHU1 BepORTHOCT8I.
? Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
tIJHKgNH C MOHOTOHHO y6blBaloiuIM npoN3BeAeHNeM ttcfLt) HepaBeHCTBO
h'CNTCH AOKa3aTenbCTBO CneAyioiuero (a}TB:nyCTb ~(O rIOno; NTenbHaR
BBIX pe3ynbT8TOB XNHtINHa,npNBeAeM HCCKOnBKO 6-OPMy1NpOBOK AOKa-
3aHHbDC NM TeopeM.B pa6oTe [27] , OTHOcfine>"ZCH x 1926 r. , coAep-
FOR 65ss-l-
'qTO6bi COCTaBHTb _npeACTaBneHNe_ 0 xapaKTepe TeopeTNKO-m4cjIO-
Inc-P-12,1'.~~g.)
Ans no' BCe* d mmeeT 6eCHOHetmoe..tINcno peureHNH B I;enbDC
%mcnax Tom N TOnTKO TorAa,KorAa paCXOANTCR NHTe-
rpan+r0t teLt) . PHA 3aKOHLIeHHEDC H3HUHMX pe3ynmTaTOB XLIHgZ-
Ha OTHOCHTCH x M?Tpwg8CK0'2 TeOpzN HeITpepbIBHEIX Ap06er4.MbI orpa-
HNLINMCH 3AeCb ~OpMynHpoBKOYip yX T81NX TeOpeM [5J H r7Ja.AyCTb
OLZ f .. ? - HenOnHble tIaCTHbie pa3JIOXern NppaIINOHanbHoro tic-
na oC 1) Henpep&BHyio Apo6b,a 3HaMeHaTe II4 IIOAXO.-
Amgvix Apo6e% aToro pa3noiceHNS3.TorAa Anil nOeTN BCeX C( C.
1 8CTBYIOT npeAemt
xaiuen II.JIeBU,' a ml- /.C TO H14 3p8HNH TeopNN BepoETHOCTe
h ~ oki
rAe C N - a6ConIoTHbie rIOCTORHHIJe / C 2,6... ;xax no3AHee
3TH TOOpeMbl MOMHO TpaKTOBaTb Kax acHMnTOT14geCKNe CBOIRCTBa AJia
CyMM tin6HOB nocneAoBaTenbHOCTeI- Cna6o 3aBMCNMbIX BenNtINH.K 3T0-
My 1Ke Kpyry NAev"i OTHOCHTCH 14 N3BeCTHb1M pe3ynbTaT XY1HtinHa noA
Ha3B3HI41eM 3aIOHa nOBTOpHoro norapNC Ma.lMeHHO B 1923 r.[0]eMy
yAanocb YTOtIHHTb oRHy ogeny gaCTOTLI paCnpeAeneI mR Hynell 14
I I 1I" Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
eAHHHI; B ABOHgHOM pa3noZeHH14 A8YiCTBHTenbHbDC clcen,xOTOpaH
(5buia pony iexa B 1914 r. XapAH H Jh1TTJIbByAOM.Ecnm nePea ~- c.,.)
0603HagHTb y}noHeHHe gHcna eAHHHLj,HaXOARigHXCJ! Ha nepBbix
MecTax pa3no:i{eHHR OT 1/2 TO,Kax OHM o6Hapyh'CHnH,Anti nomTH
Bcex gzcen /-(n)= 0 (16.n.. ) .B pa6oTe U10] yAanocb Aoxa-
nucen d HMeeT MBCTO paBeHCTBO
n-~ ~o. 2 .. Q.-.Q.r. vv
3aTb, vTO 3Ty OqeHxy iozi o 3aM8HHTb Ha 6onee TotiHylo:AnR non-
TM BCeX gHCen (n) = 0 ( n,Q,.Qr..~, .tlepea rOA nosBHnacb CTa-
TbR [15] B IOTOpOM XHHzINH TpaKTOBan 3Ty 3aAagy,xax 3aAany
TeOpMH BeposITHOCTe i.B TepMHHaX TeopIN gicen mu MOxeM chop.-
MynHpOBaTb 3TOT OHOHgaTenbHbIVi pe3yJIBTaT TaK:AnsI WORTH BCSX,
3T0 paBBHCTBO H COCTaBnfieT 3HaMeHMTBT 3aKOH UOBTOpHOrO n0-
rapmffva,xOTOpoMy no3AHee 6BIJIO UOCB IUCHO 6OnbmOe mcnO npe*-
BOCXOAHBDC MccneAOBaHHYi MHOrHX ygeHbDC?R XOgy CeNtqac HanOM-
HHTb n1mb 06 OAHOM pe3ynbTaTe,yTOtiHsIIOIgeM 3aEOH nOBTOpHOrO
norapM[ Ma B Ayxe nepBo 143 npMBeAeHHMD:X MHO U TeopeM Teopald.
riCen.3TOT pe3ynbTaT 6WI nonyueH yace s Hagane copoKOBbDC ro-
at)
AOB 3pgemeM H 1)83JlePOM Ha OCHOB npegmecTBosaBmeli padOTbt
Lr.neTpoBcxoro,noCBfineHHOK rparni i bmi 3aAagaM 4 Anx ypaBHe-
HHR TennonpoBOAHOCTM.
- - - - - - - - - - - - .
Erdt3s:P.,On the law of the iterated logarithm,Ann.of
Math. , 43, 419-436,1942 . .
Feller W. ,The general form of. the.. so-celled law of.. the
iterated logsrithm,Trans.Amerjath.Soo. 54,8 3,373-40%,1943
Petrowety I.,Zua ereten Randwertaufgabe'der W rmelei-
tungagleichung,Compos,Vath.1,383-419,1935.
~- ~ Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4 ?'~~ "
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Bonpoc MOhZHO IIOCTaBNTb Tax:HaA-TH Bce Te knM7414N (P[r-) Anfi KO-
TO L1X HepaBeHCTBO
BbIIIOJIHIieTCR AJLH WITH Bcex gxcen of. IIp1 Bcex h. 3a nCxnO-
ceHNeM .6blTb MOXeT KOHetIHoro NX tI4Cna.H3 pe3ynbT8Ta XUIHmma
BbiTeiaeT nmmb,tlTO yCnOBNO
(f (V%)
N
ti 7 2r~ QaAQoa w
AOCTaTOtzHO,a yCJIOBNe
yet"' >
00 Qi.e>w
HeO6XOANMO.Heo6XoANMoe z gocTaTotm.oe yCnoBYIe,KOTO-
pOMy AOnzHa- yAOBneTBOPRTb ( J3HKI(NR COCTONT B CXOAHMOCTN
HHTerpana _ (t)
(t) e z dt ,
A
EenN rOBOpNTB o6 NccneAoBaHNRX XNHtIL1Ha B 06naCTN HeMBTpN-
necxix 3aRatl T8OPNN gi4cen,TO B nepByIO ogepeAL cneAyeT yxa3aTb
Ha ero pa6OTM nO TeopNN AVIOf aHTOBbix upN6nIxceHmA N Ha TeopeMy
o cno eHNN nOCneAOBaTenbHOCT81d IjenbLX tmcen [53] , C97], l1c
3Ta IIOCneAHRR TeOpeMa COCTOIIT B CnegyIOII(eM:ITYCTL nOcneAO-
BaTenbHOCTE HaTypanbHETX timcen, yt.%) - gMCno CHOB 3TOH nOCne-
AOBaTeJibHOCTH,KOTOpBIe He IIpeBOCXOART 1'L IIa3OBeM rinOTHOCTTID.
rIOCneAOBaTenbHOCTN ITI N 0603HatINM Qepe3 HNEHIOIO rpaab t -
cen CymMO nocneAOBaTenbHOCTeId ,,,,? '.3RaawRaeT-
CA nOCneAOBaTenbHOCTb tINcen (f, + (L ?...+ Cf,c ,rAe KaaAoe tf~ eCTb
nI6O Hynb,nL6o tINCno nocneAoBaTenbHOCTN i(c (11. ask) XNH* II
AOKa3an,gTO ecni
Approved For Release 2009/07/09 :CIA-RDP8OT00246AO11700340001-4
 Approved For Release 2009/07/09: CIA~-.'RDP80TOO246AO11700340001-4
k
-
A ciIyqas nocne,TJ,OBaTenBHOCTetII C paBHUMN nnOTHOCTRMN.Ony6nIxO
BaHNe 3TOro pe3ynbTaTa npiBnexno BHNMaHNe MHOrHx MaTeMaTNKOB IH
`Horoqi4CneHH',.Ae nOnb1TKN -41TB
BBI3BanO ~pacnpocTpaHm ero Ha nocneAo
-
1-
BaTenbHOCTH B pa3HB1ML nnOTHOCTi MN.Aonroe BpeMR,oAHaxo,rrpo6neMa
He noAAaBanacb yCMnNRM.% nnmb B 1942 r. MaHHy a gepea roA ApTH-
Hy N Mepxy yAanocb HatTN nonHoeCpeuIeHHe.
CpeAN AOCTN} eHM% XNHMNHa B TeopLN ANOWTOBI1x npi6nm emxI
yxazeM Ha npNHaAne catgNH eMy Ba-zHLI npIHLNrI nepeaoca,xoTOpI2
CBR3brBaeT pemeHNe nIHeL HbIX HepaBeHCTB B I;enblx ZINCnax c ANO(aH-
TOBbId4 rrpN6nN)KeHIr HM K03NHW1CHTOB annpoxc4MNpyIOIQNx nNHer4HTI C
lopM.roBopsi o pa6oTax XNHtiMHa B o6naCTH T9opNN gl4cen,Henb3n
He ynOMRHyTb rrpesocxoAHbie nonynRpabte xHNr4, HanrcaHHbla 'MM B
pa3fM Hble roAbi.CpeAN HNX R XOTen 69 oco6o OTMeTNTB He6onbmie
KHH) ISN [73] H l.. ],nepeBeAeHHb1e Ha MHorve R3b1XB MNpa.
I{aK HN 3HaLINTBTIeH BKnaA XNHmzHa B Teopm bunt I "i N TeopIib
gzcen,Bce se OCHOBHaH ero ponb B nporpecce MaTeMaTNxi CBR3aHa
IQ T.-
c Teopmettk BepOFITHOCTeM.OTTonIHyBmmcb nepaoHaLianbH aAa1 CBR-
3aHHbIX C Teop1eYi uucen /3axoH nOBTOpHOro norapz Ma/ N Teopmu
IyHKI Ht /CXOANMOCTB. pRAOB N3 He3aBNCHMBIX cnytia,-AHbIX Benmg'H/,
OH nOCTeneHHO BxnIOgaJI B op6NTy CB014X IHHTepeCOB Bce 6onbmid x
6onbmmR xpyr npo6neM TeOpwI BepORTHOCTeLBOne8 TOro OH np1BneX
Y. pa3pa6oTxe ee npo6neM MHOrNx Mono= MOCICOBCKHX MaTeMaTMXOB,
IIOnO) MB TeM CaMb1M Hatlano MOCXOBCEO
BcneA 3a pa6oTaMN,nocBsmeHHJMN 3axoxy nOBTOpxoro norapi a
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR O C i vWL U g ONLY
-10
N cy?MMNpoBaHNlo paAoB co cnygaYiHbU'AN gneHaMH,nocneAOBang HCCnO-*
AOBaHPl$ XNHt1Ha,nocBRieHHb1B xnaccxcecic ripo(5neMaM C1MMHpoBa*
HHSq He3aBNCHMMFbIX cnyearmb C BenNgN4peAN nOnyeeBrn1x pe 3y3IhTaTOB
R XOTen 6bt pMMMME BMAen1Tb HCKn1OtMTenbHO rlp03pagHOe yCnOBH8
rlpMMeHNMOCTH 3aKOHa 6onbulnx xmcen B. cnygae He3R [BHCHMUX OAH
HaEOBO pacnpeAeneHrn x cnaraeMbIX,KOTOpOe CBOANTCR K CyIQeCTBO
B3HNIO f{OHegHOrO M3TCM8TNtIeCKOrO OiKNA8HNR3 r44] .Ranee HfBCHO OT-'
McTNTb IIJIOAOTBOpHOe UGHRTHe OTHOCHTenbHOg y6TOUig1BOCTH Cytw
[7d],KOTOpoe OKa3anOCb B CaMOY[ 6JIH3KOLl CBR314 C - 0opMynHposxo1'
OxOHSaTenMHNX yCJIOBHI CXOANMOCTN HopMNpoBaHHhDC Cy= He3aBHCH-
MbIX cnaraeMbnc x HopManbHOMy pacnpeAeneHNlo.AnR cnygaR oANHaxo-
BO pacrlpeAeneHHbix cnaraeiabnc XNHtWHy yAanocb O HOBpe1eHHO a
n.JIeBN N B.4~ennepOM N H83aBNCNMO OT HHX HaIITN HeoftoANMMe x
AOCTaTOtIHLIe yCJIOBNR CXOANMOCTN IC HO ManbHOMy 3aKOHy r79J.
ACO60 Hy7RHO OTMeTNTB pa6oTBI 147] H I4$J,KOTOpble MOXHO Cg1TaTb
HaganoM COBpeMeHHO npo6neMaTNKH "6OJIbm1X OTKn0HeHNlWln.K 3TOMy
h'Ce xpyry HAeA MM AOn}icHbt OTHeCTH rIOCTpoeHHe XHHMIHbnS 061gell Teo-
pwv rlpeAeJmHbnc pacnpeAeneHNH AM CyMM He3aBNCNMb1X Cnyeav"cHbIX
Ben1gNH [91] .OCHOBHoe npeAnoieHHe pa3BNTO i HM TeopIPI MoaeT
6TITb c~opMyn1poBaxo Tax:xnacc npeAenbHbrx pacnpeAeneHHVI Ana
CyMM He3aBHCNMbrx,6eCxOHeqHo Manmx Cnygar4HuC Be3INti1H COBnaAaeT
c KnaccoM 6e3rpaHNgno Aen4Mbnc pacnpeAeneHxR*AoKa3aTenbCTB0
3Toro ~aiTa, a Tame Apyr1X npeAno ceH14 TeOp1N cyMMNpoBarnR
noTpe6oBano pa3BHTNr'1 N np1BeAcHNR B Hop=O4 eopIH 6e3rpaBH .
J
Ho Aen1MBIX pacnpeAeneH14t ,He3a.onro nepeA T8M BBeAeHHUIc B pac-
CMOTpeHHB BpYHO A8 $HHeTTN N A.H,0nM0BOpOaw
Teop1R CyMMNpOBa'HNR Tp1xAbt BAOXHOBnRna XHHQNHa Ha HaIIHca-
HHe MoHorpacTi14LHepBaH Moxorpa~HR Ha 3Ty TeMy 6bma mm H3AaHa
r
B 1927 r. [35J nocne Toro xaK OH npogen B MocxoBCxoM yH14Bepc1.. I
FUR
i L~;..u .o 1.1,x'
Ammrnvorl I-nr DoIonco')(V1Q/(171(1Q ? ('IA_PfDQr)Tn (V)ARAr lI7rV Ar)r) IA I:.
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
0~.v
FOR
TBTe cne$Nanbmw- KypC Ha 3Ty TeMy.BTOpaR Moxorpac Iai 651 CBR-
3ana Knacc cecxie npo6neMu C1I4poBaHIR. C TeopNelI MapxoBcxxx
npolgeccoB N 1ccneAOBanNiIMN He3aAOnrO AO Toro 3axoHtleBHbTW
A.H.KomMoroposbnc N M.r.neTpoacxIM.TpeTbs MoHorpaNR [92] Aa-
Bana CTp0Yinoe N3noxeHNe 06U X IIpe=nb$LIX TeOpeu Anfi DWM
CyMM He3aBHCI4MrIx cnanaerfbix x NX npIMeHeHNe x xnacc ecxo t aa-
Aave 0 CXOANMOCTH HOpMNpOBaHHBIX CyMM K HOpManbHOMy 3axoHy.
C03AaHNio 3TOui KHNrI Taxxe IIpeAineCTBOBano tITeHNe cnegianbHoro
xypca B MOCKOBCKOM yHNBepci1CTeTe03TOT xppc npiBnex TorAa.K
Teoprm CyMMNposaHYIR NHTepecbt A.A.Bo6poBa,A.A.PaRxoBa,a TaKoe
MONO _
B He110CpeACTBeHHO i CBR3LI C Teopweii CyMMNpOBaHNFI HaxoAnTCH
pa60TbI XNHtMHa no apmNeTNKe 3a1OHOB pacnpe neHHR,B KOTOpbIX
YICCneAyIOTCR BOnPOCbI Iffi~IIIS IIpeACTaBneHNFI pacnpeAeneHVIk B
BNAe KOMII03L4 I14 /npON3BeA?HNSi/ pacnpegeneHNiL.CpeAN IIOJIyceHHuIX
XNH%mHL1M pe3ynbTaTOB OTMeTNM cneAyioigIe:Ka-nAoe pacnpeAenenze
cneAHeMy npeAnoxce rnro np4MblxaMT pa6oTH M.r0 Kpev"IHa no npoAonzelg
pa3naraeTCR B npON3BeAeHNe 6e3rpaHNtnio AenNMOrO K cxog erocR
IIpON3BeAeHHR xoHetmog Nnn C4eTHO nocnegoBaTenbHOCTH Hepa3no-
xIMbnc pacnpeAeneHNLI;AeneHNe ~Iyx.
IV N N
x npHHHM8IOtINX aHaideHNA B npOCTpBHOT-
?
{~ (~ ("~' COOTBeTCTBeHHO x noOAeAOBSTeAb-
sa7c (~O/ Sye)! y010 0
HOCTb BeAxtiHH ... P CO SH8cIeHNAMx 8 (F, SF) , npxgeM
npx saAaHHHX = H l?e.~= ycAOBHOe. pacnpeAeJleHxe AAM
N Q
nape 8aAaeTCA K8K H He 88Bxox? OT
AaJIbHe2uHX c.B@AeHNf O ~K~ LK~ LK B npeAHAYtxe MOM8HTH
~
CHfOBBHHOM A it ABAAmqyiwcA Mepofd MA 9'07 F npx (AB- ._:i
H8AB x n828XOHyLO CDYH 88HHBA Gol ~e, t, A), %0e Y ~ f f F
- -c S xaMepHMyio OTHOCH TexbH o S ye x F npx IB-
A E S
CTpaHCTSO (F) SF ), H88HB86MOe npOCTpaHCTBOM COCTOAHIIA Ba-
OTpBHCTBB x (~Se; S too) ~ - cHrHOROB Ha 3X0-
,Re N BiixOAe B. MD60ff MOMeHT Bp@MeHw, 8 Taize xauepHMoe npo-
VO so F)
0
Q0(~o, fo,A) = S o
TBKNO, ATO nepexoAeaa 4SyHBIMNX
ye Ha
CTBymT Yepb o (A) H8 (F,S
e
~~ Lf n A E S y AH'UB n TOM tIaOTHOM Cuy'ae, NOM S Oyate-
(PYHKLUID COOTBeTCTBymuuero nepeAaeouAero. ycTpo2cTSa Q (b,A)
n
np1JIOZH-
MOCTH ABJIAeTCA B TeopeMe IlleHHOHB TO, ' LITO MBTOAN KOANPOB8HHA
H AeKOANpOBBHNA, CyIgeCTBOBBHRe KOTOpNX rapaHTNpyeTCA BEnox-
HAHH 6M . 7CJIOBNA (18), no BNAHMOM7 He186e3H0 CBA88HH C K OARpO
Y
BAHM6M 6OAbworo KOJ1HLZeOTBa RHi opMauUH, KaK eARHOro 1 AJloro,
R n08TOMy Hew86eXHO CJIOZHE (npM'6M OJIOAHOC Tb RX B08paOT8eT
npN, WC i ) . B OBABH 0 3THM OLIBHb 8aMaHLIR80A, XOTX
nOKB LZTO N OTABJIeHHOA KBX6TOA. nepcneKTNBa noJly'ieHHA KOJIN-
LIeCTBeHHNX 04eHOR MRHHMBJIbHO2 CJIOZHOCTN.aJIrOpMTMOB OnTRMBJIb-
Horo KOARpOBBHHA N AeZOAHpOBBHNA (op.pa6oTy [31 ), rAe ne-
A06HEe OI;eHp npilyva?TCA AJ1A BJIrop1TMOB Bl !OJINTOXbHOrO
BHBJINBa). U aTOtl TOWN 8peHMA OLZeHS 8H? p60HN pa6oTB [987,
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
&FM USE ONLY
24.
N #28], rAe AeJIaLOTCH OL;eHIN Csepxy AAA cpeAHero uHcila ,AeiCT-
BMA B aJlropHTMaX AexoAHpOBBHwH AAa npoCTeAMHX. CHTyagHa.
246. HanxHaa BBOAHTb nOHATNR, HOnOAb3y SMUG B MaT8M8TH,
uecxo1 ~opMyAxpo.Bxe np$MOrO yTBepZAeHII TeopeMH WeHHOH8, mw
H88OBeM nocaeAOBaTeXbHOCTb nap C.ytaafHHX BBAHtiMH (3t,
f = t,2,...,
NH 0 M8 NOHHO- C TORtMB OR, eCJuNO< .l ( 6 *) r`t __~~?+,~h
TBKHM, t[TO npH AF060N h BeJHtiiHH z.t,...J..) N
CBA8aHk1 OTpe8KOM KBHUJB
AAHHH ri npH HeKOTOpOM HBtlaabHOM
pac npeAesieHHH AAA C OC TOAHHA KBHaJia, H QTO
H C Cho ~~)
(27)
rAe HHXHAA rp8 Hb 6epeTCA no aceM npoguccaM L.J = .., ,.
\ N ti
TBKHY, TO npH JIx)6OM Y% seiiii IHHji C~+t,... Vl J N Cat, , > ~
06p88YIDT 0006[geH14e C h HOKa?H6Hg&Mid. BHBOA p88eHOTBa (26)
COAepxKTOA AAA K8H8JIOB C KOHetIHOA naMATb? no XHH'414Hy-48AH-
CTetHy B L991 H [341. 11po6JleMa yCTaHOB.AeHHA 06[glX yOJIOBH$
AAA. ero Bh1nOJIHBHHA, 8 TBKXe ege nO%ITH COBCeM HeM3y'I8HH8S
npo6iieua ycTaHOBJieHHH yolloBHA ARE pSa?HCTBa (27) noBHAH![oMy.
TeCHO OBA88Hd 0 npo6aed8MH ~XQ nOAytteHHA yOJOBHA HH Op-
M814HOHHOR yCTOAtiHBOCTH OOOTBeTOTBy[0IgHX I{8HaJIa H 00061Q?HHa?
ice) + ' l ~T.- 1ry
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
2.12. TIpwBeAeHHBe oprMyJKpoBKH TeopeMti meHHOH8 OTpaza
IDT MBTemaTHMecim pe8JIbHyIO BapTNHy nepeAa$N Hw4opMagxH, AAA
tgeACR HeKOTOpbf3 603bWOfl, HO KOHeuHbtff OTpe8OK BpeMeHK. AO
B '60JIbWNHCTBe apeZWAypix p860T BBOANAaCb NHaA MaTeMaTHceO-
K8R MAGB,ANS84MA, npH KOTOpO$ nepeAa9a KH(I'Opma$IH MiCJzNTOR
AnKlgeflcn 6e0KOHetIHO AOJIrO. MEI He 6yAeM nPHBOANTb COOTBOTCT-'
ByIDtgxe onpeAeJleHMR, no0KOnbxy AJIR npoC Temxx cilynaes OHM
AaHH B XOPOWO 08800THHX pa60TaX, a B o61geM c nytaae rpoMOaA- 1
EN K OOAepraaTOA B (213, paaA.
AJIR npoCTeAmox cnytaaeB, yze x8y,4eHHHX B AKTepaType
paHee, nepexOA OT Teop8MFi IOHHOH8 B upMBeAeHHOL BHme ()OpMy-
J1xpoBKe K cooTBeTCTByIaIe$ TeopeMe AAR 6ecKOHetaHO$ nepeAataH
nOgTO TpwBxaneH,' oAHaoO B 06u eM cAytae STO yze He TBK.
OnMmeM HBrARAHO npHMep, paS"RCHRIOIlqNff BO8HNKaIDtgxe npo BTOM
TpyAH00TN. TIyCTb nOAJlexatgee nepeAaue Coo6DgeHNe 1t=
rAe -- o < a - Cnytae2H8R Benotawua C Henpep2BHHM p8c-
npeAe]IeHxeM, x YCJIOBNG TotIHOCT0 BOCnpONSBeAeHNA. RBJffieTOH
yCJIOBxeM nOJIH0.rO B00npOx8BeAAHWR. CKOpoCTb C08A8HKR TaKOrO
ooo61geHNR 6eCKOHetaHa. OAHaKO Cyu eCTByIOT MQTOAH KOAKPOBSHN&
N AeKOANpoBBHMA, no8Bomomxe ne p6AaTb ero uepes nepeAaIDtgee
yCTPOICTBO C CKOJb'):yroAHO MaJOf HeHyAeBOA CpeAHei nponyoK-
H0 CnOCO6HOCTbID. AAA aToro, B8RB. MOHOTOHHyIO nOCJIeAOB8T8Ab-
HOCTb MOMOHTOB BpeMBHM Tl-"-? c 88 oTpe soa E'1`n+a,TJ
nepejgaARM CBOAeHNR 0 0 TOtaHOCTbID AO
~ . Bu6paa Th?, AOCTBTOtaHO 6OAbuKM, 8TO MO$HO
OAGABTb CO CKOJTh yr0AHO' M8JIO BepOATHOCTbID OWK6K0. K 4D6O-
My MOMeHTy BpeMeHM yse AO OTOr0 MOMAHTS H8 BEIXOAe 6y.rT
nOAytLeHH OB0AOHB8 0 ft CO OKOJIb yroAHo 60AbUOl. TO' HOO TbD
K Maiolt BepOaTHOOTbIO omxdKH. TaINM o6paaou, SAGO j? >C
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
39.
X
V
a nepe,Aatla 88 6e0KOHetIHOe BpeMA TOM He NeHee BOBMOZH$. Moz-
HO nPNBeCTN TaKze (Cyu4eoTBeHHO 602ee cJiOXHHA) np&Mep 8H8-
JIorz'IHOA CHTyagHIl, rAe B KOTOOOM
KaBeTCA NHTO peCHIM pa aO6pa TbCA AO K OHI(8 B STOM Spy 're
BOnp000B. ) JIA 8Toro, 6BTb MOZ8T, nOHaAO6NTCR HeOKOJIbKO pe-
t OPMNpOBBTb onpeAeAeHNe BHTpOnMN H npOnyCKHOA cnoco6HOCTM
(Cp. P88JIDltui a OnpeAeiIBHNA OKOpOCTH nepeABAH WF4 OpMaLHN
B [53])?
3. Buumo I@HNe nPOnyCKHOA Cn0006HOCT'N V -BHTpOfHH
K onTHMBJibHI x pecnpeAejieHzI*
3.1. fOCi{OJibi{y pemeHNe npo6JieMS IIIeHHOHa AaeTOA tiepe8 npo-11 nycxHy? CnO006HOCTb nepeAamlgero yCTpOflCTBa N 1n/ -BHTponrID
00061geHNA, B8YHO Hay 4HTbCFi B} tlHCJIATb HBHO 8TH aeJINtIwHH 8
KOHKpeTHHX cJiyvaaX. KpOMe Toro HHTepeCH8 TeCHO CBAB8HHOA
C npeAHA 19MM Bonpoc 0 BftHCJieHMN_onnTHM8JIbHI x pacnge,Qe7ieHNA
H8 BXOAe nepeAaioigero CTpOACTBa ....(T.e. pacnpaAeJieHNA P a
npoCTpBHCTBe CNrHaJiOB H8 BXOAe TaKHX9 'aTO eCJIN HMeei.
pecnpeAexeHNe P N nap8 CBH88Hd uepeAa14u14M yCT?
pOACTBOM ((,~,v) ), TO
(28)
H OfTHMaJIbHHX pacnp AeJIeHNA. AJIA COO6ll eH'HH (T.e. COBMeCTHiX
pac npeAeJieHHA K(. ?~) B npON88eAeHNN nPOCTp8HOTB 8Ha1QeHN$
CoO61geHHA H8 BXOAO N Ha BHXOAe TBHMX, 'ITO naps ( ,' )
o pacnpe,AeiieHNeM 'r(., .~ o6paayIDT ooo $i eHMS (P W) I.
~( f, i) = H(a.W).
(29)
B8zH0CTb OnTHMBAbHHX pacnpeAeJieHN9 06"80HABT011 T$M,,IITO NS
HepaBOHOTBa X15) BNAHO, 'TO eo3IN. aeJI14"Ell
I 'A
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Ofir} Q rTj A
it
aaASIDT cnoc o6 nepeAaux C006U6HNA CPS W) no nepeAalongeMy
ycTpoAeTBy Q,V) , TO
H(P,W) (30)
noACKa8HBaeT KSK AOJIBeH 6UTb BN6paH MOTOR nepeAa x. -C TOtIKI
OTC[o,4a CAeAyeT, 'ITO xorAa BHTponwA Coo6n eHKA 6Axaxe x
nponyCKHOf CnOCO6HOCTx nepeAalolgero.yCTpoACTBB (Tee* 8T8
npOnyCKHBA C130006HOCTb MCn0Ab8yeTCA nOt4TW nOJIHOCTbIO), M@TO-
AH KOAxpOBaHMA K AeKOAxpOBaHws H8AO BH6xp8Tb Tex, XIT06H
p83HOCTx H(P,W) it C (n,V) - 96>A>I<
M8AB. OTCI0Aa,'e0T6CTBeHH0, CAex8Tb B61BOA, LITO pacnpeAene-
Hxe cxrHaAB H8 BXOAe I x aHatleHMA C0o6tgeHAA -fI ff 6Ax8Kx K
OnTXMBAbHHM, K SHBLIMT, 3H8'IeHxe OnTHMBJIbHBX pacnpeAeJeHUR
V Bp0HHA MBTOM8TMK8 SiTBePEA@Hxe 0 TOM, tITO x8 MaJpOTH p83HO-
~
CTeA ~3c ) - H(p,W) x C (a,V) -'i(`&.-) cieAyeT 6AxaocT
CooTBeTCTByIOI4MX pacnpeJ eJleHMA, Aajlexo He O''IeBLAH8, nOCKOAb-
Ky npOCTp8HCTB0 pacnpeAeJleHMA He KOMnaKTHO, 8 MHt OpM8I>(MA
He HenpepIJBHUA ()3HKL(4OH8A. KBZeTOA TOM He MeHee, t'ITO 8TO
BepHO B AOCTBTOLIHO OIHPOKMX npeAn0JozeH1AX.
3.2.,B 000TBOTCTBMW C onpeA@ABHxeM nponycICHO> Cn0006HO
CTM (16) 88Aat18 ee BH'IHCA@HM ABAABTCA SBAataeA 0 BHjIMCJIBHxx
MBKCWMyMB c)yHKL 14OH8A8 (a B CAy'IBe AMCKpeTHOro npOCTpOHCTBB
y -JYHXAKX HOCKOAb1mX nepeMAHHHX). le CneLU4 14K8 CB988HB
0 000611M 8H83114T4td8CKMM BWAOM P8CCM8TPMB8eMoro c)yHKItMOHBni,
n08BOAAI0t1WM ynPOCTATb pemeHxe, 8 TBKze 0 Tem. tITO H8 06-
ABCTb x8MeHeHMA ero apryMeHTa nOJIO OHH OrpaHxtl@HWA Txna He-
pBB$HCTB (orpaHxtleHAe V x Tpe608aHxe HeOTpMgaTeAbH.OCTA
paonpeAeneHM> Bep0ATHoc?eA). 06198A Teopxx B8pI8L4MOHHHX S8-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
C~fT
11~d' li f~~ a
WiLi
36.
Aati C TBKNMH orpaHHtaeHHsIMH etge HeAOCT8TOtIHO peap86OTaHa
(CM. [5] ).
nOJIyt4NTb RBHO6 BHpazeHwe AIR npOnyCKHOA Cn0006HOOTH
yAaeTCR JIHQIb B NCKJItocMTenbHHX CJlytdasX, OAHBKO B p860TOX MOH-
HOH8 N ocodeHHO MyporM ( (803, (61J, X627) . paapa6opaH anro-
pITM, CBOAi qKg SBA8Qy K peweHHIO TpaHCIjeHA@HTHoro ypaBHeHHA.
OTM NcCJIeAOBaHHFI npoBeAeHH Ha ypoBHe "5NSNtIeOKOA OTporOCTA"
N xexeTCfi BaJ WM (B 0006eHHOCTN AIR 6ozee CJIozHoro H@ANO-
KpeTHoro CJIytiaSi) BaBepwMTb. NX, pas06paB noApo6HO. BOnpOCH Cy-
tnecTBoBaHMsI, HCCJIeAOBBB oco6be CJIytIa1 N T.n. B pa6oTe McHHO-
H8 [82] AaHB Alfa ANCxpeTHoro cxytiaR reoM@TpwtiecR8R TpBKTOB-
K8 SaA84H N, B g8CTHOCTN, nOK88aH0, tiTO MHOKeCT80 OnTHMBJib-
HUX pacnpeAeJIeHNA BHnyxJIO. HHTepecHO COSA8Tb McTOAb BHtaNCJIIe-
HNfi KpaAHNX TOtlex 8TOr0 MHo eCTBa, 8 Tame nepeHeCTN NCOJIe-
AOBBHNO Ha o6ulNA CnytlaA.
AHBJiorMgHHe npO6JleMI BCTSIOT B CBHBN C BdtiwCJI@HNeM. BH-
TpOn1H COo61I4eHNA N OnTHM8JIbHUX pacnpeAexeHNA AAR 00061geHNH.
O,gHBKO SA@Cb He CAeJI3HO etge nOgTN HNtIerO (eoJIN He O4NTBTb -
6eraux SaMetaaHHA B [853).
3.3. A AR CJiyttafi, KOI' a nepeASM96e yCTPOACTBO RBIReTCa
OTpe8 KOM K8 H8JI8 6 ea n8M$1TH HeT py. HO nOKa 88 Tb (CM. f213 MIN
0333 VTO CpeAN OIITHMBJIbHUX paCnpeAeneHNA H8 BXOA@ NM@@?-
CA pacnpeAeneHNe C He8BBNCNMkIMN KOMnOHeHTBMH. OTC?Aa Jlerxo
B1 BOAHTCR, 'ITO eciw Cr, -nponyoRHaR CnO006HOCTb OTpe8KB
AINHH to , TO
Ch_hCs (31).
OAHaxo BHZ;NCJI@HNe' BeJIN'INHH Cs, RBJI5eTC5 06BtLHO Tax 3e ?pyA-
HOA SaAaaet. AIR rayocosoxoro xaHaJla des RBMRT}I 0.. GAAHTHB-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
rte-o
viz
HUM myMOM BepHa ()OPM YA MeH HOHB
- h t,03 Ps
(32)
8 OnTHMajbHOe paCnpe) eJIeHNe ABJIAeTCR rayCCOBCKHM.
AHaJiorwgHO, AJIA coo n eHHA C nOKOMnOH6HTHUM yCJIOBAeM
TO' HOCTX II He88BYICHMUMA KOMnoHeHTBMM cpeAN OrITYIMaJ1bHUX p8C-
npeAeAaHxK HMeIOTCFI. (CM. [211 ) pacnpeAeJieHHq, Ana KoTOpux
nap hi C7f ,%~ He88BHC14MU. OTCIOAB CHOBB CJIeAyeT, 'ITO eoxx
HM - BHTpOUNA, 0o0614eH%A C h KOMUGHeHTaMR, TO
(33)
HN = n Hi
Oco6eHHo npocTOf ABJIAeTCA cneAyiou a$I opMyna AAA OAHOpOAHOrO
rayoCOBCxoro CoO61geHMA c Y0AOBxeM orpaHXUeHHOCTH cpeAHe
KBaApaTHHHOt Ouu6KH
(34)
rite Pd - cpeAHe-KBBApaTH'Ha OMK6K8, 8
nOHBHTU C OOfteHR H8 BXOAe.
Pl-
- AucnepcwA KOM-
3.4. B 6oAee o6ti x CHTy84MAX ABHUX J)opMyA AJIR nponyCK-
HUX CnOCO6HOCTe3t H W - BHTpOnmri O6UtIHO He yAaeTCR 0 B CBA8A
0 BTMM CTBBHTCA HBCKOJIbEO 60Aee npOCTBA 88JPua BU9HCJI6HHA
CpeAHe# nponycKHOA Cnoco6HOCTO C 0 CKOPOCTH Cos,ILBHHA c006-
ageHHA H (cp. pa8A. 2.11).
IIpo6JIema BUUMCJIeHMA BeJINgXH C o H yze H8mA8 AOCTB-
TO4HO nOJIHOe pemeHwe AAA Ba Hegmxx C np0KJIBAHOA TOMM ape-
Hog rayCCOBC1XX nepeAamii(wx yCTpOaCTB H C0061geHM2. Aeio D
TOM, 1 TO K8K HBTpyAHO nOK88aTb (CM. rI0HCKep C ul )
B BTXX oaygasx Cpeui OnTMMBAbHUX pacripeAeJIeH14L BoerAa sorb
rayoooBCKHe pacnpeAeAeHHR. A Amt raycooBOxHx paonpeAefeHxa
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
VCR U' % 11, -
3%.
MOXHO ABTb (CM. [801,, L40J) npOCTOe BHpaXeHNe AJI$ NH1opMa-
9NR B KOHetIHOMepHOM CJiymae, AOnyCHSIDu ee 0606140HRE N Ha 6e O-
HOHetlHOMepHHA Cnytaat (B 'I80 THOCTH B [40] p88BNT M6TOA BHtaRO-
JIBHMA HH~OPM9L MN AJIR HeKOTOPBX xJIB000B raycCOBCKNx npogec-
COB). O u e c opMynbl AAA CKO OCTH nepeAa'R HF4opMaLLMH, CpeA-
H@)4 npOnyCKHO> CnoC06HOCTN, CKOPOOTH C08AaHNA 0006I46HNA tae-
pea C OOTBB66TCTByIC1gMe CnelTpanbHae IIJIOTHOCTR nOJlytaeHbl IIHHOKe-
pOM ['lOt aCTNLIHO ero pe3ynbTBTI nOBTOpeHbl B C/43. McTOAH
pemeHNA nOA06HHX 88Aata TeCHO CB988Hb1 C 8H8JINTHtaeCKNMR M6TO-
ABMH KOppeJIALj4OHHOA TeopHN CT8$NOHa'pHHX. npOL;eCCOB..
Haw6onee nOny3ApHO# B TeXHMtaeCKNX npHAOX6HNAX ABJIAOTCA
CAOAy10148A 4ODMyJIa IIIeHHOHB, opraHRta6CKN BomeAmaA B MbBmJIOHMS
paANOHHZeHepoB: EcAm CT - nponycxH8I CnoC06HOCTb nepeAa?Ige-
ro yCTPOACTB8 C KOHet1HOI nOJIOCOl nponyoxaHRA N 6enbl i raycco
BCKRM 8AAHTHBHBM IyMOM AAA nepeAauw Ba Bp@MA T , TO npN
C T ,,, W T& ~ 4 t
IJIA ApyrHx KJI80008 npot%eccoB Aejzo 06CTONT xyze. Tax, H8npH-
Mep, AAA LIaCTO xcnOJIb8yeMHX B paAHOTeXHNxe peileeBCKNX np0-
L(eccoB (8TH npoL>;ecCbi BOSH14Ka10T npM nPOXOAAeHNN y3KOnOAOCHHC
rayccoBCxMx npoueccoB taepee AeTexTOpb1 AAA BEIAeneHHA. orN6a1o-
Igeg (CM. Hanp. 1163 ) npo6nema npHBOANT cpaay x Cepbe3HBM
TpyAHOCTAM (KOe-tiTO n0 8TOM. nOBO CAeJ1 HO B [621 ).
3.5. Jlerxo BHBOAHTCA nPOCTaH s)opMyna CKOpOCTI co8AaHRA
000614eHNA C yCJOBNeM nOJIHOrO BOCnpOH8BeAeHNA, ecu! npOl%eOO
H8 BXOAe ABJIAeTCA OAHOp0AHO t KOHetaHOL& 4enbiO MapioBa (OM-L601.,
L13] ). OAHaKO, eCuw TOabKO 88MeHNTb 8AeCb uenb MapKOea Ha
npOgeCc, ASJAIO1414fl0A ( HKL Hei4 OT 46nM MSpioBa, 8aAatla OTaiO
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
XX1
Mil
FOR . ug f hwil. 16,i..W.Y..
BNTCA H@MBMepHMO CAOEHee. Ee OCTpoyMH0e pemeHNe npeAAOzeHO
BAeKBejAOM [61 . Hat6oAee aKTyaJIbHHM KaLeTCA aAecb Bonpoo
o TOM, HeAb8A AM 06o61QrTb 8TO pemeHNe HS ciyca), xorAa ycAo-
BMe TOtIHOCT}1 ABAReTCA npOH8BOAbHuM nOKOMnOHeHTHBM MAN aAAN-
TMBHUM yCJIOBNeM. BoaMozHO, uT0 aHaJIOrHt1HHe MBTOAH MOEHO npN-
MOHHTb K eue COBCBM HenCCAeAOBaHHOR aaAatle 0 HBX0EAeHNx cpeA
Heft nponyoxHOR Cnoc06HOCTH K$HSJI8 C KOHeRHOA n8MATbM. B BTOM,
cilytiae cpeAM onTMM8AbHux npolkeecOB Ha BXOAe (T.e. npoueccoB
AAA KOTOph1X AOCTNraeTCA BepXHAA rpBHb B [26-1 6YAYT no-
BNANMOMy (CM. [8 ]) npoueccH, ABuIou HecH ~yHKIUMAMN OT Map-
KOBCKMX npogeccoB. K8Z@TCA MHTepeCHHM ABTb MBTOA AAA BUiINC-
AeHNA napaMeTpOB TBKMX npoL eCcoB.
3.6. TpyAHOCTb H8XOEAeHNA ABHHX BHpBZeHMi AAA W -aH-
TpOnxA N nponyCKHHX cnoco6HOCTe4 eCTeCTBeHHO npIBOANT K MHC-
AN NCnOAb8OB8Tb MBAOCTb HBKOTOpUX napaMeTpOB K8HOAS N 0006-
u eHNJI.
AAA cAyuas, KorAa nepeuamtgee yCTpO CTBO TBKOBO,. tITO
CNI'HSA. HB BHXOAe M8JI0 OTAMtIa eTCH OT cxrHaJIa H8 BXOAe 14 AAA
OAytIaa, KorAB coo6tgeHNe TBKOB0, tITO BHaueHNe erO H8 BHXOAe
6JIN88o x ero 8H8fleHMm Ha BXOAe, 8aAat18 oK88HBaeTCA TeCHO
CBA88HH0A C HeBepOATHOCTHOI Teopmeg # -BHTpOnxt x # -eM-
KooTel /am- [56], A06-2). Apyrmm noAe8HUM or.paHNtleHNeM MoZeT
CJIyaCNTb npeAn0A08eHMe 0 TOM, 'ITO KaH8A ABJI.!IeTCA KIH8JIOM 00-
OAyvaAHUM n8p8M8TpOM. ECTeoTBeRHO npeAnoJOEXTb, tiTO B 8T0M
oiy'IBe CpeAHAH nponycxHaH CnO006HOCTb
C . Ste, S ( FPO
rAe ~(?) - paonpeAeAeHNe napBMeTpa 3(cQ)~ A (fie
F DRY
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
FOR 63 1
CKOpoCTb nepeAaux rH lopmauxx AAR nape npogeocoB (3?, S 8)
W Np
TBKMX, UTO ) N ,.i S?`1) CBA88HH COOTB$TCTBylfli48K
yCA0BHUM XBH83OM C nep8MeTpOM ,P , a BepxHRR rpaHb 6ep0TOR
no BOOM npogeCCam H8 BXOAe { 3N , AAA 'KOTOp JX oyuueCTByeT
npoi eCC H8 BXOAe?t s, 3 Ta KOA, TO need CSa~? . S?~ ~3~ ~..'~ S
OBA88Hm HccJeAyeMIM KOH8AOM , K8 pa6oT I(I6ay oBa ( E1011,C10PZ
C103]) CABAyeT, tITO ( OpMyAa (36) nosbo 3 e.T BdtaMCAATb np0ny0
KHy10 CnOCO6HOCTb mHpoxoro KA8CC8 ~HBMtlecKw-peaJIbHHX K8H8A03
pacnpOoTpaHeHxA paAWOBOAH, B tBOTHOCTA npH HBJMMM (fAi0KTy8-
ux# 45a8d. B pa6oTe 8BTOp8 [181 ()opmyJI8 (36) AoxaaaHa AAA
oAH0r0 tiwCTHoro CAytias, npxtiem AOKa8aTeJIbOTBO JlerxO pacnpo-
CTpBHAeTCR H8 0614NA CJIytMB2, Axmb 6H np0CTp8HCTBO 13 8H8tIe-
HHA napaMeTpa 6HA0 KOHetIHHN. IIOCKOAbKy OAH8KO B ynOMAHyTHX
npmxoxeH-1AX (opMy.IIH (36) n prXOAHTCA xmeTb Aexo C BeJ1WWIxHOA
wmelougeA HenpepiiBHOe pacnpeAexeHle, HHTepeCHO BHBOOTH
j()CiV 4)opmyJy (36) x AJIA aToro o.yt;ax.
OTmeTHM O.I HaKO, TO KOH8A CO CAytiaAHHM napameTpoM ABJIL-
eTCSi Heapr0AK4eoKHm npx A1o6o9 pasyMHOA TpaKTOBKG BToro Tep-
MMHB X nO8TOMy AAA Hero HOBepH8 npAMBA Te opema IJeHHOHa. aTO
He JixmaeT OAH8KO 4)H81tleCK0ro CMHCA8 8HeReHHe C , noCKOJIbxy
ero MOXHO TPBKTOB8Tb K8K npM6JIK OHH08 8H8tieHxe AJIA ' nponyCK-
HOA Cn0006HOCTM KaHaAa C meAAOHH0 meHAEDILMMCA napameTpom, K.
KOTOpum npAmaA Teopema IIIeHHOHa npxmeHnMa.
3.7. 148 BOZHHX, Ho He paccMaTpwBanmMxcK BAeCb HanpaBAe-
HMA HCCABAOBaHrA, ' OTMOTMM p860Td no BE1tdHCAeHHIO nponyCKHOA
Cn0006HOCTM MHorOAytIBBJ x KBHBAOB (65.1 x no CT8Tx0TH ieOKO$
ol;eHxe TeopeThKO-II4OpM84XOHHdx nepaMeTpOB 'pe8AbHk x K8HOA0B
H Oo06lEeHMA ( C41, 01919 157] ) .
Jtl
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4 `
FOR i LY
4
4. MCC.ueAOBaHHKe OflTHMBJIbHHX HO,QOB. 4.1. ECTeCTBeHHHM
06pBSOM BOTBOT BOnPOC KaK HBR60JIee npOCTHM 06p88oM noCT-
pOHTb McTOAH KOAHpOBBHHS N ,Aexo,AMpo3aHHS, CyIleCTBOBaHI3e KO-
Toplx rapaHTApyeT Teopema McHHOHa. MW PBCCMOTpWM BTy npo6Ae-
My Jfl uIb ,AAR Hax6oitee. npocToro oxycafi, xorAa coo61geHHe RBAA-
eTCA coo6u eHNeM C yCJIOBNeM rIOJIHOro BOQnpON8BeA8HAA TBKHM,
ZITO np0CTp8HCTB0 X 8Ha eHNtt 0006tgeHNA H8 BXO,Ae 000TOHT
N8 S 8AeMeHToB E,,..., ES ,8 PP (E~ 3 TO
co061geHAe MH 6yAeM 0608Ha'IaTb C P, TaxoA cxygaA
00060 BBZeH, nOC.KOJIbKy XOA AOKa88T@AbCTBa TeopeMH IDeHHOHa
AAA npOLI8BOAbHOrO 0006U 6HHA (CM. [211 ) noAcKa8JiBaeT, 'aTO
pemeHHe npo6JIemk AAA SToro tacTHoro CAy48A BOAAeT K8X He-
npemeHHaA COCTBBJIAIOII;BA B ee o6lgee pemeHixe. MU He 6yAeM ocTa
H8BAHB8TbCA 8AeCb H8 NHTepeCHOA cepim xccne,y0B8HHt (CM. HA-
npur.ep Ev%3 %, nOCBHU46HHO2 T8 K H88HBaemlM Hepa BHOMePHIJM KO-
ABM, KOTopie MOXHO NHTpenpeTHpoBaTb K8K HCCJIBAOBBHWA a eK-
THBHHX MBTOAOB KOAHpOBBHHA AAA KHOr0 KJIBCCa coo6u eHM .
4.2. IIyCTb 8aABHo nepeAaiolgee yCTp0 CTBO ( Q)V) . 1qe-
pea e(Q)V,19) 6yAeM 0608H8tI8Tb HBUmeHbmee # TBKOe, cTO
coo61geHNe Cp) 7 #) M0Z T 61iTb nepeASHO Llepes nepeAatOtgee
yCTpOfOTBO CQ,V). noCKOJIbRy nOCJIeJ OBaTeJIbHOCTb coo6u eHMA
rni S ABxReTCA npw S -?O NH~0pM81;K0HHO-yCTOAtiYIBOA C OH-
Tponmeg eot s , TO eCAN
4. (37)
t -+ 00 tt CCQ,V
H AAA noCAeAoBaTeAbHOCTN C Cu yAOBJIeTBOp5fOTCS npeA-
n0A08eHHH npsMOA TeOpemH IIIeHHOH8, BHnOAH@HO COOTHOmeHHO
tA, rA a W,
~ 4OO RwJIA "` .F ,,
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
(48)
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
M U,A&2' LS SLY
4%.
148 06p8THOI TeOpeMII WeHHOHa Bb1TeicaeT, '1T0 60AN H8060pOT
~?~ >
c (Q ,V")
e 4 e(Q )Vt,S,,-) >O.
(39)
(40)
t --*.CPO
NCnOJIbSyf! QOCTpOeHWA, npHM@HSBMue npH AoK88aTeAbCTBe
HpAMOA TOOpeMH McHHOHO, HeTpy,IHO nOBa8aTb, aTo eonN.noexe-
.AOBaTeJIbHOCTb (Q', V HH40pMa1 MOHHO-yCTO$%IHBB, TO (40)
MOZHO 88MOHHTb H8 6OAee CHJIbHOe paaeHCTBO (41)
V(41)
-+ad
Pa3Axuxe MexAy (PopMyJlxpoBKOMH (40) H (41) noAuepxiBaJtoab
BOAb ;OBHL(eM ( [973 ).
4.3. ECAr p8BOHCTBO (41) AOCTBTOLIHO i c ep8HfaloitxM o6pa-
80M OFMCIBBeT 80HMnTOTHKy e(?~?~.) npH yCAOBHH (38)~
TO HaO6OPOT, B CnytIae BHIIOAHORMH (37) XO4eTC$I YTOLIHHTb COOT-
Ho.meHHe (38), OgeHHB CKOpOCTb CTpeMJIOHHH e(?, ?~ -~ K HyAID.
Taiai OL eHKB OCO6eHHO B8ZH8, eCnH OTpe8OK AnH-
h
.
HH N oAHOpoAHoro KBHOJIa, nOOKOJIbKy eCAH e (Q" , V s ho)
AOCT8TO' HO M8JI0S TO KOAxpOBBHxe AA9 nepeAa'x nO AJIHTeAbHO
AeACTByzo1qeMy K8HOJIY MCXHO npOB0AHTb nO OTA8AbHHMrr..6AOKau
AAHHH ho (cp. 8H8JI0rML1Hl~le nooTpOeHHH B [333 0 1141 ) H
8H8LIHT, '40M McHbU16 Vlo , TOM npotge 6yAeT noAo6HH2 ueTOA
KOAHPOBBHHH. IIOBHAHMOMy, npH OueHbWHpOKHX npeAn0A0seHHKX
H8 (37) CAeAyeT, cTO npH HOKOTOpOY Q > 0
c, c CQ *0)
(42)
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
48.
F-- Bo BC$IXOM CAytiae, npocnezmB8A AOK888T@JIBCTBO npHM0A
Teopeid McHHOHO (CM. [213) , Aerz o 8aMeTHTb, %ITO 9T0 6yAeT
Tax, eonw npeAn0A086HHe HH~OpM84HOHHOL1 yOTOW WBOCTM nocze-
AOBB TeJIbHOCTYI (cat, ~/ t) 88MeHIITb H8 6o iee OAJIbHO6 npeA-
nOJIOXeHHe 0 TOM, TO CyUQ6OTByOT n0CJI@AOB8T@JIbHOCTb nap
(t , ~t) , OBH88HHHX nepeAaKDf MM yCTpO OTBOM (c~ , V9
TaKBH, tiTO npw JII06oM
# "*O
c(atvt)
C(Q;V. 3)
AAH KaHaJI8 6ea naMHTH ycnoBxe (43) cileAyeT, eciu npeA-
nOJIOKHTb, 4TO OIITYIM3JIbHOe paonpeAeiieHAe H8 BXOAe HBJIH@TC$
MAOCTBTOtLHO XOPOUAM? 148 M8BOCTHHX T8opeM 0 60JIbIHX yKJIOHe-
HHAX CyMM He88BMCMMIX B6JIHt1HH (cM. (.151 ), (TBKHM nyTeM
IPatHCTeIH [32) AOK888A yTBepzAeHxe (42) AASi oAHOp0AHwx KO-
He'1HHX R8HSJIOB 6e8 fSMfiTA. AAA KSH8JIOB 0 KOHegHHM tIN0AOM
C OCTOAHHR, yTBepffiAeHMe (43) MoxeT 6HTb, nOBNAMMOMy, BeBeAe-
HO MS HSBeCTHHX OL(eHOK 6OJIbWHX yKJIOH@HHA AJIS OyMK BeJINtIHH,
OBSSSHH1 X B I(enb MapxoBa, OAHB"KO B 60Aee o6u e 1 CITy84HN Hys-
HB OTAeibHBe MCCJIeA088HHH..
4.4. HpeAnOJIOzMM Tenepb, LITO ( ( n V h)" - OTpe808 ARK-1
HH h oAHOpoAHOro KBH828. TI0jio AM AaAe@ S" E S L
YcJIOBAe (37) oa auaeT Tenepb, ttTO
NBC ? (44)
rAe C - cpeAHSB nponycIHBH cn0006HOCTb KQHaAa. HccAeAo-
BBHNH, npoBeA@HHBe B npoCTeAmKx OAy'aiX, n08BOJIS OT HBAeHTb-
OA9 TO tipH' r 'i ono
H HexOTOpbx 4,, 4 >O
~~fp+T~ ?nt N qA~
n ) u '~.i'.r t*a
-- Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
H HeKOPOpOM A(E) >O
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
)()(VI
NA
e ( jv; rtnHJ)
noApe aymeBa eT, taTO
(45)
61, X4y~ fin" x-= ?O ) . EOJiee TOtIHy1O BCNMnTOTHRy
N1 so O w N-+-po ~h -
hQ 2.
(3AeCb O6O8H84eHHe Xr
C2"H7~ ' ~h~ Q
e (Q" V"
(46)
rAe C -KOHCT8HT8, nOJIytIHTb BW,ANMO H@B03MOEHO (70TH TO
M He AOR888HO nOKB XITO HM ,AJIH O,AHoro ciiytaeS) . Ae.IO 8 TOM,
WO npM NCCJI8AOB8HNx BePOHTHOCTH 4e (-, -, ? ) C TaKOA BNOO-
KOR TO'IHOCTbID CyigeCTB@HH61MH OK8861B8IOTCH BpN McTNtIeCKHe
CBOACTBB tINCJIe h .
KBxeTCH HHTepeCHHM ,AOK888Tb cygeOTBOBBHNe.KOHCTBHT
Q N to , HO euue BBXHee, KOHetaHO, HBytawTbCH NX BLI'&NOJIBTb.
BaJIOCb, tTO A RA TBKNX K8H8JL08
$ = $o ~ I2 (~) t ('(- ~.) 2u ~ w- (~~ -f- Co Ar
a=
3Ta aaA2ga OR83HB8eTCH Tpy,i[H0ik .Aare AJIH npocTeftmxx ROHeuHHX
KBHBJIOB 6e8 naMHTN. ABTOP p8CCMOTpeJI 8Ty aaAatIy AJIH R8HBJIOB
6e8 naMHTH TBKNX, tTO M8TpWL(8 BepOHTHOCTe1 nepeXOAa QJ=NI)
06JI8A8eT CJIe,AyIOIQHM CBORCTBOM CNMMeTpITnOCTN: K876,A8a es
OTpOKB HBJIHeTOH nepeCTaHOBKO I JIIO602 ,ApyroL CTpOKK H KBEABA
CTO36eg HBJIHeTCH nepeCTBHOBK02 JI?6oro Apyroro CTOJi648. Oxe-
rAe
,I P.
Approved For Release 2009/07/09: CIA-RDP80T00246A0 11700340001-4
(48)
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
. Lj
M - ICAO aJBMOHTOB B npOCTpBHCTBe CHrH8JIOB H8 BXOAe,
8 .~ - B npOCTp8HCTBe CNrH8JIOB Ha BHXOAe, eCAN
gaCTHHM cJiyLlaem p8CCMOTpeHHBX K8/H8JIOB RBAReTCR CHMMeTpNQ-_
HEIR 6MH8PHHA K8H8A C MSTPHL40A oft $4 +qis =S .
)[AR aToro KaHana 3Jia1ac [267,[273 NSYUHA HHTepecylouyio HBO
aaAaXIy '-/. B cJiyciae H 4 Hcti;t YAaeTCR noAyI9KTb JIHIIIb H8c0B
naAaioqNe BepXHkOIO N HHZHIOBO o1(e.HKy AAA 8 . 3aAaaa- TO HOrO
sunzcABHIH OKa3HBaeTCR SABCb ABEe AAR npoCTeItuero cNN-
meTPH H0r0 6NH8pH0r0 K8H828 O'eHb TpyAHO> K ee pemeHNe, BN-
ANMO,iBO8MOXH0JINmb C npiBAeta@HNeM K8KNX-AN60 HOBHX KAeL
3JIai ac 0261 HCClieAOB8JI TaKZe CJiyt8g 6MHapHOrO KaH8JI800
CTMP 8HHem, T.8. K8H8JIa C MBTpHIAe ( q"t 0 %' .. ') q, tgs~1
0 5-1 If-
He RBARIOIQeroCR CHMMBTPM HHM B H8m8M CMHCJIe. KaXeTCR, SLTO
ycVl~i BTOT peSyJbTBT 3A82Ca MOT 6UTb 06061UeH Ha H@KOTOPUR IM BOO
_ He6MH8pHHX KaH8A0B. O.1 HaKO yze AAA HBCNMMeTpNLIHOrO 6HHapHO-
XX ro KSHaAa SaAagga Haxoz eHNR nepaMeTpoB Q N CTaHO-
BNTCR OHOBB OtleHb TPYAHOA N QOAXOAH K H GA nORB UTO He BNAHH
(rpy6o rOBOpR .ThM Hciu_+ coBnaAaeT CO cpeAHeA npOnyOKHOA
CnO006HOCTbE C , TaK tITO TpyAHOCTH 8AeCb Te ze, 'ITO N
AAR CHMMBTpH'IeCKO ro 6HHapHoro K8H8A8 npN H' H c. t. ). B O6-
I4eM CAyLIae KOHe'IHOrO K8HaA8 66S n8MATH yAaeTCA nOAytITb ANmb
HBKOTOpBG 0$BHKN (CM. [831 ).
IGHHOH ,6361 NCCAeAOBaA saAaqy AAR BazHoro KABCCa rayc-
COBCKNX KBHBAOB 6ea n8MRT14 C B,IL HPHBHHIM myMOM, rAe OHO CH0-
XXx
K KO eft a oW.CM~R. v.AA Q OIAAWSO'fHO
B8 H8?A8 n0AH0e pemeHHe ANmb AAR H >., Hct,'f . HOSCHO, H80-
KOAbIO ero peSyJIbTBTB AOnyCRBIOT 06o61QeHNR H8 NHH9 ray000BC-
KNe R8 HBAH. ITOBRA14MOMy peSy JIbTB Tb1, noAy'qeHHHe AAR ONMMeTpNII-
PC-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
46.
HHX KOHetZHHX K8HSAOB 6e8 naMATH MoryT 6HTb 0006t4eHN H8 HO-
KOTOPHe K8H8JId C HenpepNBHHM MHOAeCTBOM CNrHaJtO8, o6pa3ym-
IQMMH OKpyEHOCTN I iI = 1 H TaKNMH9 IQTO NX nepe.XOAHue Be-
POATHOCTN NHBBpMRHTHH ripe nOBOpOTBX (TBKNe KBHBAH MoryT
XOpOmO OnICHBBTb nepeAet&N npm nOMODUH (Pa8OBOLt MOAyJ1$U W11.
4.5. Bo BCeX $NTWpOBBHHHX pa60TaX AAA nOCTpoeHNA MOTO--
A08 KOANPOB8HNA C HBNMeHbme1 OMM6K02 WCIIOJIb8yeTCA MOTOR
CJIyLIBkHorO KO)NpOBBHNA. COCTONT OH B CJIe,QyIOIT 8M. "ByAeM p8C-
CMBTp$88Tb ANmb Hep8HAOMN3NPOB8HHOe ROAMpOBBHNe. B cooiaeT-
CTBNN CO CA@JI8HHHMN BHme 8DMe'IaHNAMN (CM. p88A. 1.10 ) -
3T0 He eCTb cyugeCTBeHHOe orpaHNLIeHNe 061UH.OCTN. I{OANpOBaHNe
MOJBHO TorAa 8aAaBaTb KOK ()yHKL WW (CO . KOAOM &.?.,(S)
MH 6yAeM Ha3HBaTb Ha6op 8H8tieHPIA (ES) . Iqepea
, (S)) MIS 6yAeM 0608HayaTb HammeHbmee # T8K06*
e ((2y
'ITO 00061geHN@ (Pjm ) MOffi@T 6HTb nep@AaHO uepea nepeAalo-
Ii;ee YCTPONCTBO ((2 V) C MCIIOJib8OB8HNeM KOANpOBBHNA &(S)N!
H@KQTOpOrO McTOA8 Ae1OANpoBBHMA. IIpe,i(IIOJIOHNM Tenepb, HTO
H8M 3aAaHO HeKOTopoe pacnpeAeJleHNe BepO THOCTOR h(. )
H8 [I OCTp8HCTBe CNrH8JIOB Ha BXOAe (X,S) N CPLCT@Ma Heaa-
BNCNMHX CJIy'iBtHHX B8J1W iHH T.~, ..., YS - c pacnpeAeneHNeM
MN 6yAeM H8 8HB8Tb (S) CJIyu8; H
KOJOM,- 8 M8TeMBTNt1@CKOe OANAaHHO
e(a v, s) _ E {e(Q,v, ms(s))}.
CpeAHeA BepOATHOCTb? OUH6KN ARE p8 CnpeA@JIBHNA _'( (. )
(49)
OUeBNAHO, tiTO F cu; V, S) C~~Vi$) . BC@ 1488@CTHNO FI 1lle-
peQKcA9HHHe BHme 80NMIITOTN'eOKNe OIdeHKK CBepxy AAA Bepo$IT-
HOCTH (Q, Vjs) OCHOBBHB H8 BTOM HepaDeHCTBe,. OAHaXO
Q
NCOAeAOBaHHe Ope,ILHOA Be9OATHOOTH OWN6KN HM@eT H C8MOOTOA-
_ Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4 ? '~`'
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR D~ J L L i- ,
44.
TeibHBA MHTepeC, IIOCROJIbxy OHS Xapa KTepw8yeT BepOATHOOTb
OIIIM6XM AAA TNn1UHOrO KOAB (ECTb OCHOBOHNA H8AOATbCA, UTO
B IUMpOKOM xiiaooe cJIyUaeB BepeH CJIeAy MA 88KOH 6OJIbWNX
QNOen: npx H -~ O?
e~~hv j (r H7~l~~
XXJ I
e /fin vh L~h"]) (Ito BepOATHOCTN) (50)
KazeTCA TBKZe MHTepeCHHM MCCAeAoB8Tb nOAp06Hee BCNMnTOTM-
UecKoe pacnpeAeJleHMe AAA e~G2~ Vh L(L 1 'J)~, np1 h-> ?O
B KaQeoTBe 'T (.) 6epyT Ualge BC8r0 OIITHMaJIbHOe pacnpeAe-
JIAHMe CMrHOAB Ha BXOA9. npw ATOM M8 o6U'gHOro AOK88aTeJIbCTBB
TeOpeMH McHHOHB (CM. 1213 ) BNAHO, LLTO 148 yCAOBI#IA (43) BH-
TexaeT He TOAbKO 000THODieHM6 (4Z)., HO N 00Jiee CMJIbHOe yT-
BepzAeHxe .
; 1 S{) = o 9-0 c' ~Q v+~ (51)
IIoBMAMMOMy, B mM OKOM iilacce CAyUae3
igh
(52)
'BHUMCAeHN 8HGUeHMA G N nOK8 JIMUIb AJIA TeX ze cJlyQaeB,
KorAa (CM. Bliule) MCCAeAOB3AMCb KOHCTBHTH Q N . OK888-
JIOHb, UTO npw H i He t; t 4 J C = g . 3T0 08HaUaeT,
UTO npM AOCTBTOUHO BHCOKMX CHOpOCTAX nepe.auw MH(~)OPMBL;Xff
cJIyycaAHIIA KOA aCHMnTOTMtleCKM (C TOUHOCTbIO AO KOHCTBHT) T88
me XOpOw, KOK K OnTNMaJabHHA, UTO nO8BOJIReT (CM. [371 )
OTpOMTb NXOpOIUMeW KOAU McTOAOM MoHTe-KapJIO?
OTMeTNM, OAH8KO, UTO AAA rayccOBCKOnO IBHBJIB IIeHHOHy (867
npi4MJ1OOb 6pBTb B KageOTBe 'Y (.) He OIITMMaAbHOe pacnpeAe-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
FORS i dA.USEONLY
4%.
JI@HNe, a HeKOTOpoe. JINWb acNMnTOTNtdeclM onTHMaJIbHoe pacnpe-
geJIQHHe (BMeCTO pacnpeAeJIBHNA II He88BNCHMHX HOpMaJIbHWX
BeJINgWH CO CpeAHHM O H AHCnepCNe> PS , eMy fl KWJIOQb
BSgTbPSBHOMepHOe pacnpe,AeAeHNe Ha PI -MepHOIL crepe peANy-
Ca. Vrn PS ) . NHTepeCHO, He uo a JIM QOA06H! npHeM npNBeoTR
K yM@HbmeHN) 6H H B ApyrMx cJ1y48Ax. npN 1-i Her: E
ABHHO BUPBX@HHA ARE C. H e Taxze H8 TH yA88TCS, HanpH-
Mep, AAR OnHC8HH1 X Bu me CHMM@TpHtleCXMX K8Hazos
G H + 2 %IE(i )+ ? ,aJf--
(53)
I4HTepeCHO, WO BO BCeX NCCJIeAOBaHHHX cJlyuaAx Hcdt;t. OK8-
88JIOCb TOtIKOA HBJIOM8 npOH8BOAHOR d;yHKIgMH g H~ nOBNAN-
XXXV
MoMy npM H - Hcr;t 6-419 , oAHaEO 8T0 He AOK8B8HO ,ASKS.
AJIA cwMMeTpwt4Horo 6NHapHOrO KaHaJia (cp. npo6Jl. ) . 3a-
Aa4a BunmCJIeHHA 0 N npon e, tleM 8aAatla BdtIRCJI@HNA KOH-
CTBHT 0- H g H KazeTCA peaJlbHJM H81 TH G N B AOCTaTOq
HO WHpOIOM KJiacc,e CJiy' aeB.
4.6. TpyAHOCTb npexTHaecxoro ocyn ecTBneHNA nepeAatt N..
TeopeTHtleCHoro NCCJI$AOB8HNA npOHSBOJIbHUX KOAOB npIBeJIN K
BbAejieHNIO oco6oro xJiacca rpynnoaux KoAoB (CM. [441, [897,
[10] , [911 ) .
npeAnonoSHM, MTO paCOMaTpMBaOTCA KOHetIHhIA K8H8JI 6,ee na-
MSATH TBROA, TITO 1IHCJIO 8JIeMBHTOB B npOCTPBHCTBe CNrH8JIOB H8
BxoAe M rAe ~, - npocToe QNcilo, a - ueJioe
tIHCAO. OTOZABCTBHM fo C np5Mb1M npoMaBe) eHMeu f ANKJIA-
tleoxHx- rpynn nOPRAK8 , a y" . 6yAeM paCeMaTpNBaTb K8K
rpynny, ABJIAlOI>;y1OCA npBMHM npoMaseAeHHeM n rpynn
KOA 1(S)H88OB8M rpynnowim, GCJIN OH o6pasyeT noArpynny rpyn-
f-
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
?g
nH y" O6o8HaLIMM Liepea a((2,V, HaMMeMbmee 8HageHHe
? ,
BepOATHOCTN e(Q,V~ A(S)) KorAa MHHIMYM 6epeTCA no
. cuuPercotb rcn4ccO j
BoeM rpynnOBHM KOABM. UKa8HBaeTCA, 4T0 AAA OAHO x'KBH8-
ABB des n8MATH, 06AaAa@Lgmx CBOiCTBOM CHMMeTPNUHOCTN nepexoA-
HOR MBTpMIgH, 8 Taicze 'AAA CHMMeTpMIIHOro 6HH8pHoro K9H8JI8 N
61TH8pHoro icaHaata CO CTHPBHHeM (CM. [273 ) npm VI-000 N
Htot; t
1% V
(GIL
H T8KHM 06pB8OM Haxxygmmg rpynnoBOR KOA BCHMnTOTHLIeCKN
CTOAb ZO XOpOm, K8K N HBNAyLImiA HX BCeX KOAOB. 3TOT ()8KT
ABAAeTCA cxeACTBHeM Toro, 'ITO nPH HBKOTOpOM eCTeCTBeHHOM
onpeAeReHHN nOHATMA c.ny'IaAHOro rpynnoBOro KoAa opeAHee 8H8-
tleHNe BepOATHOCTH omw6KH no BCeM rpynnoB61M KOABM BCMMIITOTA-
Aecxm coBnaAaeT npH BCeX H C. (Q n., V ~ C)-? H3)
TaKmm o6pasoM, no'ITH Bce rpyn.nOBHe KOAH npw H > Hest
yCTpoeHH K8K OnTHMaAbHHA KOA (t4T06H yTORHHTb BTO yTBepBAe-
HNe, XOT@JIOCb 6H H AAA rpynno.BHx KOAOB nOJIyt HTb p8B@HCTBO
xX- xV! TNn8 (50 ),. 11pO 1H i Hvut a0HMIITOTH%ieCKOe nOBeABHHe
((i^ V~~ [QhHJJ HeM8BeCTHO Aaae AAA 6HHapHoro CHMMe-
TpMH@CKOrO K8H8JI8. PesynbTBTH, OnwCaHHHe BHWe, MOKHO nOBN-
AHMOMy pBCnpOCTpaHHTb Ha HeOKOJIb80 6OAe@ mIpOKHA KJIacc Ka-
XXXIZ)
HBAOB 6e8 n8MATH (Cp. np06A. ~CXY~~ ) . OZIeHb HHTepeCHHM K8-
zeTCA BOnpOC 0 TOM, 6yAyT AN rpynnoBHe KOAH BCHMnTOTN'IeCKH
OnTHMBAbHHMN, ROTA 6H AAA npOCTeAmxx KJia0008 K8HBJIOB c na-
MaTbm.
LO8AaHHe OnTHM8JIbHHX a4re6paMUeOXXX McToAOB KOANpOBe-
HNA AAA M $ , nOBHANMOMY, HeBOSMOSHO. LITO K8c8eTOA K8-
H8JI08 C HenpephBHhM MHo eoTBOM OOCTOAHNL, TO BAeOb, nOBNAN -
_MOMy, BO3MOSHB 06o6n eHNa H8 ORY198 1, ynONSHyTHI B 081180 0
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
npo6A. X3d, KorA8 MoZHO oTOzAeCTBNTb c rpynnoA no
yMHOgeHMtO KOMnA@KCHHX XIMCeA (I I =1. . IIYT} K 06064eHMIO
aJlre6paxtaeCK1x M@TOAOB KOANpOBSHNA, npH KOTOpOM OHDI CTBAR
6N npMM@HIMH, CK8XeM, x rayCCOBCKMM K8HSA8M 6e8 naMATI 3
HBCToRi ee BpeMA T8KZe H@ BWAHO.
4,7. 3aAatla noCTpOeHHA OnTNMBJIbHUX KOAOB, T.e. 8aA848
C08,rAaHMA OTHOCHT@AbHO np00 THX BArOpHTMOB AAA 88A8HWA KOAS
di(s) Taloro, LiTO eCc~'; V ",ACS)) COBnaAaeT (xi no
KpBAHeA Mepe) 6ANsKa K e (a' v~ S) OZIBHb TpyAHB. EO yAP-
eTCs pemHTb ANWb AAA OTAeAbHHx He6OAbMHX 8H8Z1eH1R M A S .
OHS CTSHOBNTC$I AIIIIb H@MHoro Aerne, ec ix OrpaHHgXBaTbCA p8C-
CMOTpeHNeM rpynnOBUX KoAoB (CM. [89) ). BBNAy aToro sseAeHH
N nAOAOTBOPHO NBytIaIOTCA NHHe H@BePOATHOCTHUO M@TO,IW AAA
[1p4chnOA016R-u N OIL;- 42' V ') - KoHtYH%'u 4&k(Qn Gt3 RA M..TW .
o4eHKH KatteCTBa KOAa. orAa y^ - aTO np0CTp8HCTB0 noCAe-
AOBaTeJIbHOCTeft =.~ ~~~--?~ ^~ , rAe tj Z nprHHMeeT 8H8-
ueHMA ~, ..., M . 110AOXMB .P ((~~,..., ~.,), (~..??. `~..
pBBHHM Z'HCAy MHAeKCOB (. =4, ?-?, to TBKMX,' ZZTO A C-47- ~JZ
MCI npeBpaTIM ~ B McTpML1ecK0e npOCTp1HCTBO.
ZMU Ha8OBeM KOAOBb1M 80CTOAHHeM KOAB {?(- IFS))
d (Vii') i wu n ~7i l~(E~)~ CEj))
(b5)
MHTyHUA noAcsaBHBaeT, ZITO, Boo6ige roBopti, AAA K8M J1OB C
CHMMBTp14eCKHMN M8TpM48MN AAA KOAOB C 6OAbIIMM KOAOBHM p80-
CTOAHxeu BepOATHOCTb OWM6KH McHbue. HeTpyAHO2 OgHaxo, npx?
B@CTN gaze AAA 6HHapHOro K8H8A8 npxnepU K0 A0B T8K)X, UTO
~ d ~ HO e(Qn V ; ,L) > e(Gt",V ; ) ?
COBnaA@HMe KOAOB onTNMBJIbHHX- B CMUCAe BepOATHOOTx OMM61N
N B CMHOA@ KOAOBoro paCOTOAHNA yAA80TCS A0Ka8aTb- AMMb s
i~ L~ 111 Ell
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4 ?
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
51.
Becbm8 ya:;oi (cM. C881 ) KAacce oxyRoeB, , xorAa cylgeoTByaT
nx0THo-ynaKOBaHHHe KOAH, T. 8. KorAB BCe npOC TpBHCTBO
11OI5HO npOACTaBNTb KaK CyMMy HenepeceKaIOtQNxci ciep 'paBHoro
paANyca (nAoTHo-ynaxoBaHHHf x0A 06paByIDT I;eHTpH 8TNx cep)
He AOK888HO AaZe 8CNMnTOTNtIeCKOe COBnaAeHNe 8TNx AByX
nOHHTMA OnTHMaAbHOCTN, XOTH MOZHO BWABNHyTb rwnOTeay, ZITO
eCAN nOAOZNTb
a,1(n, S) = rna~t dl C2 Cs~) (56)
XLJ
(59)
IIpo6Aeml, aHafOrxt1Hue npo6AeMaM M, N dt BCT1MOT T8KZS
lll N He HBXOAHT peMGHNA, eciu orpaHHtloTbCA AN?b NSy'IBHNeM
~1 V rpynnoBFJx KoAOB.
HOHJiTNe KOAOBoro p8CCTOAHHA MOZeT 6I Tb eOTeCTBeHHHM
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
no B06M KOA8M 4(S) 06 "e.a 05 AAA nepeAa$N A]INH8I h ,
TO N8 TOrO, tiTO AAA nOCA8AOB8TeAbHOCTN KOAOB , (C2''~1),
h. t,2,...
J (A! Sn H7) 41
~ICn.E~~""3)
BEITeK8eT, WO.
e Can V P%) J (EfHJ)
( M ---I- o0
O6p8TH0e y TBepZAeHRe HeBepHO. OyHKIUHR
(58)
CI(v+, S) B~ItI>acAe-
H8 ANIb AAR HBKOTOpbix SHateHMA apryMeHTO3 (CM. H4npuhep.C?1,Cfl]
N AO CNx nOp He pemeHB, HBCMOTpR H8 8AeMOHTapH0CTb ( OpM7-
ANpOBKH, Hao6oxee eCTeCTBeHH8H 8aAB'Ia 0 BHLINCA6HNN KOHCTBH-
Td d TaKOA, tiTO
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
fox u9 h L bt LT .
06p88oM Bae eHO N ARA npOCTeImMX KSH890B 0 HenpepBBH1 M
MHOZeCTBOM COCTOAHMA. Tom B K8ueOTBe H8AO 6p8Tb
06I HOe eBKJINAOBO paCCTOAHHe N xccJIeAoBaHNe. 0I(h, S) C6ne-
88eTOA C HSBeCTHHMN reOMeTpIUe0KNMH WCCJIeAOB8HHAMW 0 8aT[OA-
HOHNN npOCTpaHCTBB Cdjep8MN.
nH
4.8. fpeAnonoffieHNe
BBeA8HHoe BIme, AB-
no3IOZeHMK 0 nOCTOAHCTBe CKOpOCTH nepeAauw. OAHaxO HexOTO-
pIR HHTepect npeACT8BJIABT NCCJIeAoB8HNe Toro, 14T0 npONCXO-
ANT N npm NHO4 acNMnTOTNKe S . OKa8aAOCb, B uaCTHOCTN,4TO
88Aau8 CyIIqeCTBeHHO ynpoigaeTCSI, eciiw npeAnoAI08;MTb, 'ITO
S c Co-+St . 3Aec H81IIJ10 pemeHNA aHaAorM npO6JIeM
yCLII (CM.EtJ;IHTepeCHO, &T0 npw 3TOM AAR 6NHapH0r0
KaHaAa oC >. , HO d npu S ), npo-
6JIeM XL1 ,)(1-111,XLty , BonpoC >D BHIINCJIeHNN KOHCTBHT C t.
3Aecb, KOHe4HO, $ > , T8K L1T0 CJlyuat HHA K0A 0Ka8k B8eT-
CA xyze.OnTKMBJIbHOrO. HO yze AAA HeCHMMBTpWt4HO 6NHapHoro
KBHaJIB 6es naMATN (He rOSOpH 0 60JIee CJI08HIIX CJIytaiX) BCNM-
nTOTNKB Q (GZ V OCTaeTCA HBNCCJIeAoB8HHOA.
npeAnonozeHNe S = CO,*s 08HatzaeT, U TO paCCMSTpA-
BaeTCA ne peAaua c OtleHb MaJIOA CKOpOCTbIO. Apyrnm KpefHNM.
CJIytIBeM ABJIAeTCA nepeA8u8 C MBKCHMBJIbHO BOSM0ZH0R OKOpO-
CTbIO, 6JIN8KO1 K np0nyCKHOR Cn0006HOCTN. M.fMHCKepOM 6wza
yKaSaHB CJIeaylougas npo6JIeMa. nyCTb Ha1160JIb1uee
8HaueHNe S TBKOe, RTO coo n eMe (P, I'm )M08eT 61 Tb
nepeAaHO uepea nepeAalolgee yCTpOROTBO HSAO
HSycwTb 8CNMnTOTNKy jr(n,#) npsi # = GoNS4, h a'4 ?
nerKO BNAeTb, %ITO a (aVo ,lr(h,#)) TeX uTO
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 _.r.I
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
HSYR6HN6 XCnle)
5a?
eCTb OAHS N8 l)opw" wcc3IeAoB8HNI BC} M-
nTOTMKM BepOATHOCTA a ((2", Vn S) ? HeTpyAHO nolaSBTb, 'iTO
ARA OAHOpOAHoro- K8H8n8 6e8 n8MAT14 C CAMMeTpxLIeCKOA M8Tp11I-
ueA,
_y2 nc -auS%rn
(60)
r,qe C - nponycKHBA CnO006HOCTb KaHana, Gr - pemeHxe ypa-
BHeHI+I 4P64E.) r{!)J't Ae ~~ ? ) - HopM8JIbHaA . yHKIWJL
paCnpeAeAeHAA, 8 (CM. (48) ). HeKOTOpb16.
--j V
OgeHKN AnH .Al'(n, #) MOSHO HBBneub 148 p860T Bonb4OBALja
LV! KHTepeCHO HCCneAoB8Tb ~(hIL) AnK ApyrH-X KaHaiOS.
Bee BBeAeHHHe Bbl?e nOCTBHOBKN SBABUN o6 BCNMITOTHUeC-
1 yh
KOM HCCneAoBaHHH e (f S) MOYHO CHIT 8Tb 88CTHUM8 Cny-
9asMH CneAylnugeL saAaux: yK88aTb oTHOCNTenIbHo n.pOCTyLo (dyHK-
L VII L[NIO $ C h, S ) TaKylO, ZITO6H npH h --~ vo
e(CanVnS)
paBHOMepxo no S (CM. aH8AOrW'iHHe 3aAayx B npeAenbHOA
TeopV5r cyivu cnytiaAHHx cnaraeMux ([527)).OAHaxo 8Ta npo6ae-
M8 CTaHBT Ou@BNAHOA JIHUIb nOCne pelueHm npo6neMu X MI
4.9. KoJMoropoBuM 6Hn yKa8a'H HBCKOnbKO IHOA B8pMaHT
nOCTBHOBOK saAau aTOro pasAena. A MMeHHO: BO N8MeH@HHS
npeAnonoxeHNA pasAena 4.1 6yAeM euNTaTb, tITO pS
rAe Q 14 5 - 4.enhle uwCna H OTOSAeCTBHM 8H8ueHHA C0o6Bqe-
HNA C Ha60p8MN ~E
~oEyo
AJIS B06X E yo E F P 'E F
(68)
3) ((,Z Y~ V ), ire ~'t ABJISMTCA OTpeaKBMH HeOAHOPOAHHX
KBH8JIOB 6e8 fBMSTH AJIHHB f , C KOHegI*IMH MHOZ@CTBaMH CH-
rHasz08, npmRem yCJIOBHS V. OTCyTCTByIOT, 8 t YHKIgHW
rAe (4', a, ...,Cl 3T0 BC@BO8MOZHHe H860pH aJlemeH-
TOB AE A H { (2a,, 4E A ,~ - HecoTOpBe BHnyKJIOe mHOxeCTBO
nep@XOAHUX 1yHKLj}! (Mu 6yAem rOBOpRTb, t4TO MHOZeCTBO nepe-
XOAHkIX (YHKLWH$ QQ, Q E A I BE1nyKJIO, ec Ix npH Qq aeA
A (,Z o , a E A , SBAaBaemwx maTpHLtaMH (q, ,
COOTB@TCTBeHHO, 4)yHKLAHS, COOTBeTCTBY101It8H maTpML O
+(t-~,~) o-c Taxze nptHaAJIeXHT
I~) Q COCTOHT Ha COHet4Horo, He3aBHCSiE erO OT t'dHCAS
IC Co061geHHI PC W~ ~~ = 1~?- H BCe nOCJIGA0B8T@JIbHO
CTH 00061leHHO (P 6 WZ') 2, ?? ?? HH4Opm814HOHHO yCTOA-
2') YCAOBHx WS He 88BHCAT OT a pacnpeAeJleHHA Ps moa-
HO OnHCBTb Kax pacnpeAeJleHHe Ha6opa P (3v)),
rAe 3 S o6paayWT KOHetiHYIO oAHOpoAHyio gerlb Mapxoaa' o ~HR-
Approved For Release 2009/07/09: CIA-RDP80TO0246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
F Ff u.!'A
5%.
CNpOB8HHNM %i1CJIOM COCTOAHNI, npNUeM Bee 83eM@HTEi M8TPK4 Be-
pOSTHOCTel nepexOAa paBHOMepH0 nO cc 60JIbile KOHCTBHTd
? t
si 4)
3 ) YCJi08NA W He 88BRC11T OT , a coo61geHNS (pe
ABJIAIOTCa HeOAHOpOAHHMN no BpeM@HN c006tIeHHAMH C noKOMnOHeHT-
HHMN yCJIOBNAMN TOUHOCTH BocnpON8B@AeHNA N HeSBBNCAMHMM KOM-
nOH@HTBMN, npNuBM Pal= PO $... $ PQ` rAe (c41..i 4S ),sE4
8TO BCeBO8MOEHHe H86opII aileMeHTOB Q C -A 14 PQi q c- A
BHnyKJioe MHOAeCTBO pac.npeAeJleHMA BepOATHOCTeA H8 KOH@t4HOM
npOCTpBHCTBe X~ ?
RpwBeAeHHHe npNMepI nOKa8HB810T, 19TO npAMaS TeOpeMS
McHHOHa BepH8 B ,IZOCT8TO4HO o6u ei 06CTaHOBKe. C Apyro# CTO-
pOHB 06lUHOCTb ee Bee me M@Hbme, ueM 06qHOCTb TeopeMH McHHO-
Ha pa3Aeiia 2.10. Tait, cc opMyJNpoBaHHOe yTBepm,AeHNe 6yAeT,
Hanp!Mep, B oo61Ae rOB OpA HeBepHO, ecJIN cpeAN c Pans) eCTb
Co061QeHNA C OAHH8KOBbMI1 Pg , HO pa8IU MN WS- (aA@Cb
A
40
FUR .%~'is +L 1)
BCe CBe,t1,eTCA K Heo6XOn$MOCTN yAO8AeTBOpWTb 60200 CNJIbHOe
yCJIOBNe TOLIHOCTi1, ABnR@igeecA nepece'LeHMeM HeCKOJIbKNx wr
a AJIA pemeHNA TaKOg 38ASUM OTBeT ,gaeTCA McT0A8MN pe8Aena 2)'?
AaAee, ecnN B npRMepe (aHanOrw'IHo 3") MHo:ecTBO
(~a ,cA j HeBI nyxno, TO AJIA TOM, XIT06H TeopeMa McHHOHS
(uktaaoE ud 4c,o (6Y))
OCT8JI8Cb aepHa HyEHO B enpeAeJIeHNN 6 SaM@HNTb CKCT@My Q`
H8 ee BHnyiiiy1O o6o2onKy.
b.4. OTMeTHM H@KOTOp8e B81HU6 icTHHe CJlycdaH 1JOPMJA
(633 N (64) ,U nponyCKHOR Cn0006HOCTN N aHTponMN CNCTeM.
nyCTb CQ Y, V11), re P CNOTeMa Bcex KBHaJIOB 6e8 naMJITN
AJINHH N C aAANTNBH1 M UIyMOM, MOI1HO0TbtO Q1yMe PN N MOIQ-
HOCTbIO CNrHBJIB PS . TorAB C Cr) SaAaeTCA 4)OpMyJIOA MGHHO-
R8 (3 2 3 . Aaaee, ecuM (aar), ye I' ONCTeMB Bcex -nepe-
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
L.
IL
59.
ABIODUHX yCTpOfCTB C KOHetIHOA nOJIOCO npOnYCKBHHA W C eAAN-
THBHUM myt4OM AAR nepeAatiH 88 BpeMR 7', ccr) - aaAaeTCR
aCNMRTOTH eoKH (OpMyJIOg I1eHHOHB (35). HaiOHeI;, ecru -
CQS~WS~ ~F~ - CHCTeMU B06X 00061lteHH4 C nOxOMnOHeHT
HUMH yOJIOBHAMM TOLIHOCTH, 8aAaIo11IHMY! OrpBHHtIeHHeM cpeAHeKB8A-
PSTNLIHOO OMH6KH KOHCTBHTOR Pj H U CnePCHAMN KOMnOHeHT 00
06U OHHA H8 BXOAe, pB BHUMH t , . TO AAR H (o) BepHO BH-
paseHHe (34). lopMyjw (32), (34), B OCO6eHHOCTH (35), Bbl-
BOAHMHe 06ERHO AAA rayc0OBCxoro cJlyuas, npWMaHAIOTCA He npa-
KTNxe B MHOrO 6onee mfpOKOM HEaooe CJiytiaea, B TOM '1HCne N'
TOrAa, KOrAO o XapalTepe pacnpeAeneHHA HHxIerO onpeAeneHHO-
r0 CK888Tb Henb8A. OnpaBAaHHeM nOA06H0n npaKTNKe MOZ P CAy-
XNTb npw80AHMas Bu me HOUR NHTpenpeTauHA STHX (DOpMMA.
5.5. OTM@THM, tlTO OCTB6TCA OTKpkTHM BOnpOC 0 CO8A8HNN
06DIHX McT0A08 AAA BHLIHCAeHHR npOnyCKHOI~ CII0006HOCTN CCt'')
ti
H BHTponHN H(b) , 8 TBxae COOTBBTOTByiouIHX ?OnTNMBnb-
HHX CTpaTerwA" (T.e. 8H84eHM % , I N , H8 WHOTOpUIX
XYTTI
AocTNraLOTCA 8KCTpeMyMU B (6V N (63) (cp.npo6n. XY
MY). J,BJIee K8AeTCR HHTepeCHHM HCCJIBAOBaTb 8 CTHAe SaABc,
p8SO6p8 HHHX B p88AeAe 4, BCHMIITOTHKy BepOATHOCTeA omH6OK RP
npH eANHHX M@TOABX nepeAaaI.
B Ka4BCTBe nepBOrO more MORHO npeAAOXHTb NCCJIe, oBaHHe
BepoSTHOCTH OMM6KH AAR CNCTeMU Bcex KBH8AOB 6ea na MATH,
M8TPHI;H KOTOPH X- ABARIOTCA KB8Ap8THHMH (HICCHpOBaHHor0 nOpRA-
Re, npMgeM HX AKSVOH8AbHU6 8JIOMOHTH Pa > 1- # , rAe # -
88ASHH8A KOHCT8HTS.
6. KOANpOBuHNe 0MCnORbSOBBHMGM AOnOnHHTenbHbiX CBeA@-
HNa. 6.1. HHorAa e0TOO TBOHHO npeAnOJOXHTb, KTO NO BxOAS
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
6?.
HaHajIa MSBeCTHH M MoryT 6HTb no3TOMy MCn0AbBOB8HH npN ROAN -
pOBBHMN HeROTOpHe AOnOAHMTenbHHe CB8A8HMA 0 BHatIeHMM COCTOK
HNA RBH8A8 MAN 0 8H8'L6HNM CNrHBAOB H8 BHXOAe s npeAHAyfgMe
MOMeHTH BpeeaeHN (nOA06HBe cBBAeHMSI Mo1beT nPHHBCTM K8H8Ji 06-
p8TH0A CBA8M MAN HBKOTOple McTOJJ BoHAMpOBBHMA opeAbi, Tapes
KOTOpyLo Be,AeTCH nepe.Aaua). BCT88T BOnpOC 0 TOM, K8K npN
BTOM M8M6HMTCH nOCTBHOBKB H pemeHMe npo6AeMH leHHOHB.. AAA
He1OTOpHX uaCTHUX CAycaeB 8TOT BOnpoc M8yT48A0A B pa6oTaX
IeHHOHB ( [813, 1841 ) N BHTOpa [1810
6.2. Ma npOCTOTH MW A8AMM nOAHyIO MBTeMBTNtiecKyzD (dop-
MyJIMpOBKy 88,A8t;N ANUib AAA cAytL88 OAHOPOAHOFO K8H8JIa C AXC-
KpeTHI M BpeMeHeM, XOTA t1MTaTenb 6ea TpyAa nOAMBT Kai 8T0
onpeAeneHNe paCnpOCTpaHAeTCA He o6ul4a CnytLafl. Me npeAnoAo-
#HM, tITO Kp0Me o6"eKTOB, BXOA$IflHX B onpeAeneHNe oTpe8K8
AANHH h Or HOpOJHOr?o H8HBA8 C AMCxpeTHBM BpeMeHeM, 8aAa-
,
Ha nocA6AOBaTeAbHOCTb MBMep1MHX npOOTp8HCTB (2 ~ Si
Ha8HBaeMHX npOCT BHCTBBMN AOnOAHNTeAbHHX CBe)eHMA B MOM@HT
KpOMe Toro BaAaHa nepeXOAHaH (IYHKL A
N N L
4 ,t , tit,A to ...It
,
4E 4
EF
4E s
E y
,
.8
i
o,
NSMepwMaa no COBOKynHOCTN nepBHX 31-2- nepeMaHHHX M AD-.
AAFD[AaACH MepOA no A , C HBrAAAHOR TOt1KN 8p8HMA # t C.,
8aA8eT pacnpeAeJeHMe BepORTHOCTe i AAA cBeAeHN11, HBANtLHHX
H8 BXOAe KBH8A8 B MOMeHT , eCJN 88A8HH allatieHHR CMrHaJOB
N COCTOAHMA KBHBAS 88 npeAHAy[gae Bperrn. TaIciM 06paaoM, npN-
BoAMMag (opMyANpoaKS BHAIOtda3eT B ce6fi N TOT c7Ly'aL, KorAs
cae LeHMR MOryT 6HTb OAytLafHHMM, HarpNMep, M8-88 nOMBX. B .KS-
HBAe 06paTHOR CBABN. 0AHBKO npM ETON npeAnOAaraeTOR, TITO
4NKOHpOBaH M HSM8MCHSH ueTOA nepeAa$M no KBHOJ19 06pOTHOR
""' Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
L It
CBSSN? -HHTepeCHoi"I K838ABCb 6M 88ASUS, B KOTOpOR CtINTBeTCS
S8AaHHuM ANmb CaM K8H8A 06p8THOR CBSSN, 8 M6TOA. nepeAaTN
no HOMy Bu6Np8eTCfi OnTMMBAbHNM. n0Ax0AoB K peweHNIO nOA06H0A
88ARKN He BNAHO?
3c-~ '~~t-1 e
YL(axeM BBSHe luI a 'I8CTHue CAytaax. ECJiN ~ = ~o X F
N ~-j~~~~'J , J& Z 1 ooopeAOTo a8HO B TOtIKe 0I'-`. TO MM
6yAeM POBOpWTb, LITO 3a,AaH ;KaHBA C noIAHuMN CBe,AeHNAMN 0 npo-
muIom. Ec,N N
~~'~ ?~ 88BHCNT ANWb OT
H OOC p@AOToueHO B COOTBeTC TBy1QIQefl TOQKe, TO Mil 6yAeM POBO-
pHTb, ITO.88AEH KaHBA C n0AHOR 06 aTHOR CBSSbIO (NHOPAB, He
COBCeM TOtIHO POBOpiiT, TYCO BAeCb HMQQTCS K8H8A 06p8THOA CBS-
SH C 6eCKOHe8HO i nponyCKHO Cn0006HOCT bIO) . Ewing HSKOHeg..
,z ftc F A 121 ~., ?~ 88 BNCH.T ANmb OT X8 K Mepa, c OC pe-
AOT0teH8 B COOTBBTCTByIOAeg TO'IKe, TO MW 6yAeM POBOPHTb, 'ITO
88A8H K8HaA C nOJHHMN CBe.AeHNFIMN 0 Orb 000TOAHNN.
IQeH 48 ( p, W 2) npw BCeX 7Q, H Bcex AE Sy XSF
eHNA (pv') 0 HCIOAbSOBBHHeM AOnOAHNTeXbHUX. CBe,AeHN
eCAN yK88aHa CNCTeMa B@JNLIHH C
~ t ... ~ - . ~ ~ CO 8HalaeHNAMN B X X F
000TBeTCTBeHHO TaKaS, 'ITO 1) n8 p8 05p83yiOT 0006-
6.3. MN 6yAeM roBOpMTb, ITO 8a,uaH Cn0006 nepeABtIH 0006
H88 BepOflTHOCTb
ti
= Q0 (1 #.:c, Yi-t, A) (70)
3) paonpeAeJleHNe. BeiiNAIMHM ?e OuBuaAaeT 0 H8ti8AbHMM peCnpe-
AeJeHNeM K8H8A8 P. , 4)p m xw6oM N JI06OM A G S~
C N N
fit, A)$
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
ft, 9'{sheA/1.,,..,t&_ , ~,,..%'~-~-?,A#,~,
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR - RFF b~AL ~~'U10MI1Y
5) npx md6oic - x 3 0 01 A cc S,
9tIIjkcrA 11,c.
6) npH JII OQM AeS
6L. -
(72)
S
(73)
NarnRAHnA CMUCJ yCJIOBHA 1), 2j, 3), 4) ace/H. YcnoBHe 5) os
HanaeT, TO npH BB60pe CNrHaJB H8 BXOAe B R ' -BA MoUeHT NO-
nOnbSyeTOA AMMb SHBHHe coo61geHKA H8 BXOAe, SHatieHHA CHrH8n8
H8 BXOAe B npeABAygNe MOMBHTH N AOnO)HHTBnbHHe CBeAeHHA
YCJIOBHe 6) 08H8iiaeT, UTO ABKOAKpOBaHHe OCHOBBBBeTCA JHUib H8
8HOHNN CNrHan8 H8 BBXOAe.
ECJIM npOCTpBHCTBB.AOnojHHTenbHHX OBeAeHMA COOTOAT
HBKAOe N8 OAHOrO BJIeMeHTB, TO 8T0 O8HatiaeT, TO AOnOnHHTenb-
HBX CBeAeHMA HeT. HeTpyAHO nOHATb, tITO B BTOM Cnygae BBeAeH-
Hoe onpeAeneHHe BKBHBaJIOHTHO onpeAeJieHHK) paBAena 1.40.
6.4. IeHHOH pBCCMOTpeB B f 843 Cny48 KaHan B 0 nOnHUMB
CBeAeHNAMN 0 COCTO$HNH TaKOrO, t1TO gio~~do,~i~ol~ 88BHCHT
JIMMb OT f 1 (ecam AOnoJIHNTenbHHX CBeAeHNII HeT, TO 8TOT K8-
Han npeBpauaeTCA B KBHan 6e8 nBMATH). OH 7K388n npxeM, nos-
BOARIOUI4A CBOANTb HCOJieAOBaHNe KaHOJIa C AOnOJHHTeAbHBMH CB8-
;ABHHRMH K 06HUHOMY KaHany. Ero peccyzAeHHK nerxo pOCnpOCTpa-
HAIOTCA H8 C8MHi 0614HR CJIytiaA. A NMOHHO: BcerAa MOXHO noC-
C11aco~a
TpONTb nepeAaiougee yCTpoACTBO TaKOe9 UTO Cyu;ecTBOB e-
AeUH 000614HiHA (p,W) tdepea 8TO yCTpOACTBO 8KBH88JIeHTHO Oy-
flO0TBOB8HMI0 Cnoco6a nepeAaUH Toro ze c0061QeHHS tiepe8 nepBo-
H8'BJIbHBA KBHan C HOnOnb8OBOHNOM AOnOJIHHT@JIbHUX oaeAeHAL.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4^T.
ti
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
68.
L 1t1
CTpOKTCA aTO nepeAaioiqee yCTpOACTBO TaK: IIpOCTp8HCTBO :1. COB
n
n8A8eT TaIC O2 K8K H B nepBOHBtiaJibHOM XaHaJie,C ABAee
BBOASTOA npOCTpaHCTBB Boex wamepiu x
oTO6paZeHwA 3 NO H8 2 e e ,( C eCT6C TBOHHHM 06-
paSOM BBOARMOA c? -aJire6poA RBMepXMHX nOAMHOZeCTB A nOlB-
rBeTCA . y = yo (e*~ x??? X ~o ~n . TO KacaeTCA nepeXOAHOA
~yHKgxx HOB oro nepej amgero yCTpO OTBB, TO B cny' ae K8H8Ji8 0
nOJIHHMR CB6A6HRAMR 0 COCTOAHRR HaAO CLIRTBTb nepeAaioigee yCT-
pOACTBO KaHa3IOPI C T8M le, LITO N paHbme np0CTpaHCTBOM 000TOA-
HRA N HOBOA nepeXOAHOA nJIOTHOCTbH
d ? HaI O, 8TO61 nepeAa4a npOKCXOARJig T8K,
88JIHCb p8BHUM11
$aK N B npeZHeM K8HaAe., Ha BXOA KOToporo nOAaH cLirHB.U
TaKKM o6pa80M, nO KpaAHeA Mepe (I0pMaJIbHO, npo6neMa McHHOHB
yCTpOACTBB, HO Tenepb He AOJIAiHO BJIRATb H8 KOARpOBaHNe) oK8-
ryT 6HTb MaT8M8TKTI6CKH onpeAeJIeHH K AJIA HOBOrO nepeAalonqero
(d& zk, N CBeA6HHA 0 npOIJIOM (KOTOpiie MO-
B TeX oiiytasx, KOrA8 H8 BXOA nepeAalouqero *yCTpo1CTBa noAaHO
MN yxaxeM JIRmb, LLTO ero OCHOBHBA HAeA OCTaeTCH TOR Ze CBMOA;
rAe (~,) 8H8'IeHHe f yHKLU I4 (~{~) npN G{k= fa
B o6uteM cJiymae on.peAeJieHRe nepexoAHoA C'yHKIRH rpomo8AKO N
~D (~(a~, g/ L 1 T ') = 90 ` a ( 0 J, ~O , OO,
AJIA KBH8Ji8 C AonoJIHKTeJibHHMK CBeAGHNAMN CBOAKTCA K npo6iteme
WOHHOH8 B cf OpMy3IRpOBKe paaAejia I. OAHaKO, n0CK0JIbKy noCTpO-
eHHOe nepeAaion>.ee yCTpoptOTBO oxa.8uBaeTOA o'eHb CJIOXHbuM, TO
K HeMy HenpMM6HRMb KpNTepwn KH'opM84HOHHOR yCTOA'NBOCTN,
o6cyZAaBwRecA B ? 2,.x nOTOMy OCTBeTOA OTKpHHTHM Bonpoc o
BNBOAe cneuoC Rgeoxox AJif aToro cxytias yCJIOBNA NH OPMaLjNOH-
-HOA yOTOAi0BOOTK.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
6.5, 08MBM NHTepeCHBM RBAR@TCA BOnpOC 0 BBc1CAeHHN.
nponyCKHOR On0006HOC TH K8HOJIOB C AOnOJIHHT@JIbHBMN CBeAeHNB-
UN. OTM@TNM npezAe BC@ro CAeAyIoIgWA ',BKT. npoaycKHaaa OnO006-
HOCTb K8HBA8 C nOAHBMN CB@A@HN$MN 0 nPOMAOM BcerAa COBnaAB-
@T C npOnyCKHOI Cn0006HOCTbLO (Toro ze) KBHaAB C nORHBMId CDO
A@HNRMN 0 ero COCTORHNx. 8TOT (PBKT HBAHeTCA CAeACTBHeM 60-
A@e o61gero pe8yJIbT8T8, KOTOpBR MH He 6yA@M npMBOAHTb 8A@Cb
no,i po6HO N KOTOpHA C HBrJIRAH0 TOt KH 8p@HNA nOK88HB8eT, 'aTO i
06paTHaA CB$8b nOAe8HB.AAR yBeANqeHMH npOnyCKHOR Cn0006H0-
CTH AXMb B TOR Mepe, B K8K0A OH8 AOCTBBAReT CBeA@Hfa 0 00-
CTORHNM KOH8A8. .ajee, NCnOAbBya McTOA, p83 BHTHi B [183,
H@TpyAHO nOK888Tb, LITO npOnyCKH8$I Cn0006HOCT-b K8HBA8 S AO-
nOAHLITeJIbHBMN CBeA@HNRMM COBnaABeT C 06HUHOA npony.OKHOA ono-#
'
C06HOCTbID 8TOr0 KBHaila, eCJIN BBnOJIH@HO CA@Ay?IQee ycxoBAe:
IIyCTb CJlyaafH8e BeJIHt14HB, CB$38HHHe nepeA8I01QNM
yCTpOACTBOM C IIpOCTPBHCTBBMN CNrHaJIOB H8 BXOAe H BHeexOA6
~-46,s-40).* 'CYo~ yo~ N nep8XOAH0 t -jyHKIUNe Qo(, 1`~ ?~ a
Torna 'J(j' He 88BAC14T OT , 8 888NCNT ANmb OT
pacnpeAeieHI$ 11pWMeHRH BBCKa88HH0e yTBBp11ieHNe K K8H8-
Ay 6e8 naM$TN, nOAytlaeM peByJlbTa T ieHHOHB 181] , rxecaluii,
RTO 06pBTHOR CBA3b He yB@JIHtWBaOT npOnyCKHOf1 Cn00o6HOCTM
KaHaia 6e8 n8MRTH (B MOM@HT Ony6AHKOBaHNR pa60TH [181, rAe
AOK838H TOT 88 peByJIbTaT, BBTOp He 6HA, x Coz8A@HNIO, SHBKOM
C pa6OTOA lOHHOHB).
I pyroA BaZHBt npMMep KOHR48, o6AaAaiolgero yK88OHHBU .
CBOROTBOM, AaIOT H8M raycOOBCKNe nepeABlotgxe yCTpOfCTBB (eoAa
npeBpaTNTb NX B KOH8A npHOMOM, OnIXCBHHHM B p88;IteAe 1). Aejo
B TOM, MTO AAA TOKHX KOH8AOB ao ~'~ Q~ '~ 3A ar rayccoaCitm
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
6?0
nepeA> out r - yCTpO1CTBO!, npMtieM OT f 88B14CET JINQIb CpeAHNe
8H8AeHNA, 8 He .BTOpbie MOMeHTbI. !48MeH8HNe ze CpeAHero SHatle-
HNA 'It He BJIHABT H8 NHI OpM8ijMI0 t(i, t ) . Tarim o6pa8oM
06p8THas CBASb He yB6JINtINBaeT npOnyCKHO1 CP0006HOCTN rayC-
COBCKOrO nepeAatoi ero yCTpotCTBa. TO 6bIAO BnepBHe OTMetleH0
M.IIHHCKepOM. MeHHOHOM C843 BH'LNCJIeHB nponycKHaA cn-0006HOCTb
AAA ynOMAHyTOrO Bu me CJiacca KBH8JIOB C nOJIHb1MN CBeAeHNAMN o
COCTOAHNN. ppyrMM HHTepeCHEIM ctpNMepoM MOryT CJIYXKTb K8H8JI1i
CO CJlytlaflH);IM napaMeTpoM H nOJIHHMN CBeAeHNAMN 0 8H8ueHNH na-
paMeTpe 6 F 1 . 3Aeob (op. (36 )), noanANMOMy, OpeAHAB npo -
nyCKHBA CnO006HOCTb.
. (74)
a
rAe Cg - opeAHAA nponycKHaA cnoco6HO07b AAA ycJlonHOro Rai_
H8318 C n8p8MeTpOM 8 . 3T8 copMyia Aerxo OR88 1BaeTCA (Cp.
AAA CAymas, -KOrAa npoCTpBHCTBO 13 -xoHetlHO. IIocxoAb-
xy OH8 HBXOAHT NHTepecHbie npWAOZeHNA (0 ee wcnOJIb8oBaHNeu
OBceeBNtleM, -IINHOKepOM H IgI 6aKOBHM nOKa8aHO, %ITO B HeKOTOpHX
c w8Ntlecxw pea3IbHHX CHTy84 MAX 06p8TH8A CBASb MOXeT B noATOpa
paaa yBeXHZIWTb nponyCKHyIO cnooo6HOCTb KaHaJI8) a cnly'zae He-
L1V ripepNBHOro pacnpeAe3IeHNA napaMeTpa, HHTepeCHO AOK888Tb es
H B 061geM CAytaae (op. pasA. 3,6). BBZHO HaytINTbCA BHtlWCAATb
L V nponycKHylo CrIO006HOCTb x8HBAOB C AOnOAHNTeAbH}MN CBeAeHNAMN
N- B ApyrHX cJytaasx.
MHTepeCHO TaKX6 HCCAeAOBBTb BOnpOC 0 TOM, MozeT AN AAA
OAHOpOAHWX K8H8AOB 6ea n8MATH NCnOAbBoB8HNe O6p8THOA.CBA8H
MH
yM@HbIIINTb OnTHMBAbHyLO BeposTHOCTb OMM610M Q CC2nP V"
(CM. 4.2). HeTpyAHO nOK88aTb, LITO AAA KSHBAOB 0 CNMMeTpMtiHH- ;:
MW MBTpN48MN npx H > H"; t NCnOAb80BaHNe OdpaTHOA OBASK
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
151
a" U1a1-al
tug tJI141Ij1~i'
66.
He MQHHeT KOHCTBHT Q -N Pa B COOTHOM6HN11 (45). B pa6oTe
BAaABCB [273 COAepzNTOR 88MetlaHme, KOTOpoe MOzeT 6UTb NHTpe-
n H -C "C.%**
npeTNpOBaHO K8K yTBep neHNe 0 TOM, UTO AR NHapHoro ONMMe-
TPXtIHOrO K8HBJI8 C O6paTHOA CBSi8bi0 OK88HBaeTOK BepHHM CO.OTHO-
uieHNe (45) C KOHCT8HT8MN Q N ? , 88AaBaeMHmM CHOB8 paBeR-
CTBBM (47). ABTOpy HemSB@CTHO AOK88aTeJIbCTBO BTOA Teopewl,
H1IA KBHSJI C CHmmeTpHtrHO2 MSTpNijeg (itaK yK888H0 8 1271, 8Ha-
LVI
JIorn4HUA (8KT Aerio riposepxeTCx AJISi CHMM8P MQHOr0 6NHapHOrO
K8H8A8 00 CTNPSHNem). OtaepeAHOA KazeTCA T8KXe 88AIRS NOCJIe-
r N nH 1
L Vu AOB8Tb e C(l V J JISI r83/000BCKNX KaHaJIO8 6e8
naMS1TH C SAAHTNBHUM 1U MOM H O6p8THOA CBgabV *
8 TBKxe TO, B KBKOA mepe OH8 paCnpOCTpaHSieTCH Ha npON8BOJIb-
7. Hoe npNmeH@HNii [IOH ATX IIIeHHOHOBCKOR TeOpMN NH(0p-
MauNN. 7.1. B npeABAY14Nx pas.Aeiax aToro o68opa mu paccma
TpHBaJIN JIHWb BOrIpOCH, BXOAsu;Ne B p8MKN C(iopMYAHPOBBHHOR B
paaAeAe 1 OCHOBHOA npO6JlemH McHHOH8 06 OnTNMa3IbHO2 nepeAa e
NH(opMaIuuH. 3Aecb mu XOTHM KpaTKO 06CyANTb .NHUe, nOKB RTO
JINIIIb eABa HBmeTNBUIMeCK, H8npaBJIBHHB npNAO$eHNA nOHHTNA aHTPO
nNN N 14H()OpM8u14N. He HCKJIiOtIeHO, TO B '6yAyugeM BOe 8TH Ha-..
npa BA@HH$i C OJIbIOTCSI B Hexyio eANHylo Te'OpHIO, H O A8 Tie KOHTypH
TSKOA eANHOA TeOp}N noxa TITO He Bm=H*
nepBHM N8 TBKHX H8np8BJIeHNA HBJIRBTCH NOnOAb8OB8HNe
o6o614eHHoA OHTponfm n8pB pacnpeAeJieHNI'S Kai Mepa xx pa8ANSNS
B 88A8tI8X MaTeM8TNqeCKOA CTBTNCTNKH. 0680p MHOrOLINCJIeHHUX
p860T H8 8Ty TeMy COAep1HTCs! B'H@AaBHeR KHNre K8i6eina [511.
B OCHOBHOM Ony6ANKOB8HHHe p860TH COAep? aT nep9' I4oJIeHHe OBOA
OTB 0606uqeHHOA aHTpOnMR, nOATBe.pzA8MIgHX, %ITO ee yAo6HO NO-
nOJIb8OB8Tb B Kat160TSO MepHOTBTHOTH;LeOKOro pa87innN8. Boned
U1 J J +~lf~ v'~ ULl.i,~
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
resHbu EBae?Ca Hem nonutisa nozea?t?b. %&TO vepe3 O0G116H8yD
3f~?pontao Aa8?Ca acuunrOTRI16e1s+R o?Ber He aesoropbe Rxeccw-
QecOee npoo&e4&b o:4;TeMayg 3ecso CTBTI CTaXA. ':AA H?38BVfC$TMiis
Hada.-ACHH9 alO C.4ea8HO a p86osaX Yypbe 1 3J w AReeafif;a ]?
i "ODOR p8 $O?C 'roopywima. 1T1iHCfi8Pe a I'apaeee yK+eeNB8e?Cil.
%3TO 0?ueuetHHe ncvmnrorw,4eCxwe pe3ynbT9Tbf p9cnpoCrp8HaUU is
ojieHb tlit19OKW MCC 8498NC1!MH?C f!CaNTBHwL.
7.3. FTOpoe uHoroooe,eee fHanpaBAeHAe npeacraBaeHO
nose 4TO ARIUb 8 " opae NCCAeaOPGIiAR c wcnonbaoBsHmeM noya?W$
3lltpon MN W3BeCT41OR N3 RVIYARpHbX KHt r no laif3Te4ifiTWhe 3RADIM
O SWAeJeHNs ~TiRbt11H8 t VoHeTW H8VXeHbTPIM IMCAGli aa8eii1+f88H'f3A.
`ffiKOfr KccaeAaeaHHe riaveTO 9 L491 x noapatif}f~ pa3 Oro a KH4iI-
re [4J.3iiecb uo&Ffo a op4iy$:fpoeaTb o61:,yao npoun(MY o .bK}fw-
MaAbHOI4 1314C8e ai(CneZ2K9eii?OF, Heo6?4;oAAmbx AAR ILOCTWaeHRR l8-
xoro-ro aii9HAR x, noBt MMGOy, OCRanTOr ECxoe pef11ef.we a?oA
npOdAeMW U096T 6HTb epu He&u?OpL ( ycnoB4 Hx JABHO %epea "OHS-
TWO allTpOnYWo
7.4.. A ?ZOp9 HenoHRTHD npgg4 HH, no ROTOPHa af;Tg onmu w 44iH~
op4raL R noRf3i7RNCb 8 HeKoropdx TeoperRnc-farpobb' OHCTpy4a-
,uwwt (cm- 1503 ). ltepcn8RTVP 1 STOro HaopaefeHW HeKCRH.
A VTOP oxe roAO pwr A. ` .r oneoropora, #.C.11viscxepe w P.M.
firRO4~l8. UocTOffHiduuR AGH?81 ? C H1 M3! BO OROrOM OTDO3149CA Ha
coAepSaH4ff4 asoR CTeTbw.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 0
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
66
JIHTEPA TY PA
1. AABaapH C.A. CpaBHeHNe OnTNMBAbHIIX CBOT CTB KPI4TepH6B
HeHMSHa-ILK p COH8 N B8JIbA8. TeOp. Bep. H
ee np!M., 1Y, 1959, 86-93.
BaKyT LA. - K TeOPHM KOppeKTNPYIOIWNX KOAOB C npoH8-
BOJIbH8M OCH0B8HMeM, H8y H * OKA. Blom.
MKOaN, PaANOT.H 8AeKTpoH., 9 1, 1959,
,99j 19S$, 1203-
co osg.~e2 ~awh , J. At f k. Vat.
9 ..t3eac.~tW A, 6,W s aI.4. C . Tko wca si cew. A-a 71+t c rant E~ e? o
c.Q.a4s of cActn", Ahn.. fk.Sl.,SO, fssg,/223-
lZy/.
10. BOpOAHH I.I. - H@KOTOpHe BOnpOCH TeopHH noCTpoeHN8 HOP-
F KOAOB, HBynH.-Te H-.06.
aANOTeXH. N 8JIeKTp. mM.A.C.lIOnoBB.
C6. TPYAOB, Bun. 2. M., 1958, 110-1514i
11, Tkt u.d(v iJ ta2 eftTo c 6 ktensik J? ~ot.KsQ -
fheoqj, hh.. .~tati? r4ti.f. X29, 9Si, $o9-S~l.
12. C4k9eSo4 L. ,Two ,'tQAtia s Oh 4kt 4 Sic fkeo u
tk fOtAtMI,oh +k" t3 S it, /?S-/Yo
13. XWHTINH A.R. - fOH$ TMe eHTP GO P~4N BepOSTHOO TeA.
Yon. MBT.H8YK, 88 I 3, 1953,.. 3-20.
3. BaXBaAOB H,C.- K Bonpocy - o titcie apN McTNLIeCKNX AeI cT-
BNA npM pemeHNN ypaBHeHNS IIyaCCOHa 9M,
KB8Ap8T8 McTOAOM KOHexIHHX paaHOCTe$.
Aoxit. AH CCCP, 113, 1957, z52-254.
4. BamapiH r. n. - 0 CTBTNCTxnecxoA O$eHKe SHTponNN nooze-.
AOBaTeAbHOCTH He88BNCNMHX CAy'iBAHHX Be-
XMIH. Teop.Bep. N ee npMM., 1Y, 1959.,
361-364.
5. Beef- w.an P. ,,p 2Anan4ic p~oAAto rr.++~i w$ P~t,~,,eet'-ow,19S7, P~tiM at-
tow 21.KivatsiE~ P.ts&t
I t~ E++{'`apt &f f %me~roN Fihije sfmie
6. geackwt~ ~ ?, ,~tsf P~
j4 a ~c &o c.4a: n s , T~ca~?s. of . I~ 9'ccs
2o t-des. (1u?c4.1 lYq.4(c(ow
coh~ I-+ Ik f ^ vq. Tl`+''mod, Sfa
1?' . , P/c*y*t, t 9s?, 13-20. LL
, c42w .~h~!IttK GO.~y `- kAtNaCItiJ~~i G/w0hw4d,
7 &
,.Anw. )'o19. S1cut..0 so, '(Cwt, 12Y2- 12YY
8. 1664wt ? Z., 13/kaiMa.u 1.?., rkem-%sia.K A.J, 04 SkAI%H4,S
+Q'e co
t~'cair, s~iisiu+, ektar'c+?~ foot fi+uto-Ste m;,6e-S
FOR 0,
For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
14. XKHtWH A.H. - 06 OCHOBHUX TeopeMaX Teopxx HH.OpMaLMH.
YCn. MaT.HOYK, 11, N4 1, 1956, 17-75.
15. Otawt ch H. S" L&4 hdu JtaM f 1.i o-rtw. - 4 K4 fc it ea
ciJ.,
pi's, 1 g
16.Obove.9, W.r, )goof W.,.? An 7 JAWJAsa,oh -tb &suL
thRO ~'Cgo~ow~ S i ?
$'M~Jl OML~ 1Cbii2~ ulr y,%o~
'cou4-o LNG(. , i G'tAW -k;tL L ee.4t Co,u,~.~,,c4 'hc.
17. A06pyMHH P,J!. - 110 nOBOA,y o MYAHPOBKI OCHOBHO1 Teope-
M~l IIIeHHOHO (Pe8I0Me AOK48.AS Ha 3aCej a-
HNx HayLIH.-HCCJIegg. QBMxHapa no Teopxx
BepoaTHOC TeM19/W 1957 r.). Teop.Bep.
x ee npNM., 2, 1957, 480-482.
18. Ao6pyMHH P.A. - Ile eAa4a NHIOpM8>UNM no K8H8JIy c 06 ST-
HOl CB 3bIO. Teop.Bep. x ee npxA., ,
1958, 395-412.
19. Ao6pymxH P.A. - YapougeHHUR MBTOA- axcnQpMMeHTBJIbHOk
ogeHKN 3HTpOnMN CTe4HOH8pHO nOCAeAo-
BaTeJIbHOCTx. Teop. Bep. x ee nplUI.,. 3,
1958, 462-464.
20. A06PYMHH P.J1. - 064a$I c opMyJIMpOBKa OOHOBHOk TeopeMN
IIIeHHOH B TeoPpNN NH 0 Mag$N. AoxA. AH
CCCP, 126, 1959,.474-477.
21. Ay6pyMHH P.L - 06u;?aa copMyANpoBKa OCHOBHOA TeopeMH
IIIeHHOHB B TeOpMN NH(-`o MaLNH. YCn.MBT.
H8yK, 14, J 6, 1959, 3-104.
22. A06pyMHH P.A. OnTHMBJIbHaSI nepeAat a 1H opM891KH no' s8-
HBAY C He1BBecTH1MN napaMeT aMN.
PaANOT. N aAeKTpOH., 4, 19599 1951-
1956. --
23. Z06PYMRH P.A. - IIpeAeAbHHA nepeXgA fOA SHaxOM MH `'.O MS-
qWN N 3HT OnIN. leop.Bep. x ee npMA.,
5, 1960, 28-37. -
24. o~oo~ a .L . E {-b't:a mkE TAc ? .%. IN?. T1i iti tT--S, jV?(
,
I S9, 3
25. AEHKHH E.B. - OCHOBBHHA Teopxx MapKOBCKNX npoLLeOCOB,
9., 1959, INaMaTrxa.
EE. s P. , Cooth~ fbt k.m CA kL"j 3kE CoN,
&c A 4, 1955, 37-46.
27. ie.'as P)
CodG'K~ 'f ft, two hloi'3a C" $4J4,
T ~ LomJo1j Ca.,OL.A lass,* t.h.11 Issb,
Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4 ---
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
78.
28. E ps 4-c; r m. A ~ .Mj &tc i c AtOcUft% ~~. a 9i eftOLAWce-
29. EpOXHH B. - # -BHTpOnMA AHCKpeTHOrO an g9AH0 0 o-1McKT8.
Teop. Bep. x ee npuA., 3, 1958, 10307.
30. IaAAeeB A.K. - K nOHATHIO 2HTpOnmx KOHeL1HOA BepOKTHO0THoL'
cxemii . Ycn. maT. Hayx, 11, P.19 1.958,
227-231.
31. ~ci ~glt i A . ?J4 Kew Bas c th o"c",4 O$ the
n s.. 7Iz , Ig54, PG T-y,2-2L
32. Fc;Ksk.4'11A et/ar.Ct BowKo(S ir- woisi c.Ita?N,At Wtfhowt
haetl~.o't~ ARE T'ca.uc. YMfrC4u. Tkso'c3 l3ss,t,
.N ? L, 113--t y
33- Fc; % 106 % .A ~ Foum Az4ioks c H ~~t w r'er f ha,p ''A/ Yi
14c G Ow- ki eQ ISsif
34. 1--cir.Ski A , oK~~cc cow'-+~ Eke5A 444 4mJ C61 CA
K'e. sc
~~c -~i hi [e - ,~ eau oK~ ' c~4va M w,2i , 3rH . aqi Cc tt,~.
35. F Ab C.: k' .A .
13 Sci, 2, . u-YV
0't &-. M,i Ee - lue k " cA q m 044 04 vt/1t ovo C.'~wr+~ -
fo, Su e4i. r, uVo2, 1359 3YS-$ S2
co-W r..,_ cam.
o% f kQ west & . k o q Q4 4 MUG( c&S cOK VCX 1'2
36. F scAear. 7 , Tkt c94i icca cc, nc f Es ~
Phas , 34, /Vis$, a23-22Y
3:. r-owfa:~+2 A .1~, pr.~P~^~o N W. w . OK "6U113
sg, tug ' , T/c".s .,A ..r
Jk. ~,nB2 .,S,7 , 1358',635'-G 4?.
38. reAbcaHA M.M., noJIMOropOB A.H., RrAom H.M.
K o6uuemy. on eAeAeHxr+o KoAUgecTBa xHC opmau1H. Aoxx. AH
CCCP,1" , 1956, 745-7486
39. I'eJb(8HA B.M? NoAmoropoB A.H., HraoM M.M. - .
KOJINLIeCTBO HH110pm84xx N BHTPOnHA AJ* -HenpepBBHex p80-
npeAeiieHMA. Tp yyAH TpeTBer0 m8 em. C eaAa,.TOM w,
M., Ha-BO AH CCCP, 1958, 300-20..
4A reAb(haHA &i HrJi0m AM. -
0 BURNOJz?HlHKOAHgeCTBa HHjOPMa4HH 0 CAy7aa.I~HOA -yHKLUxx,
COAepnau~eAcsi B ApyrOA TBKO 4)yHKuHH. Ycn. daT.HeyK, 12,
3 it 1957, 3-52.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
41. rHeAeHKO B.B., KojiMoropoB A.H. -
lIpe~ eilbHUe pacnpeAeneHHA AJIA CyMM HesaBHCRMHX BeJIHt1RH.
M. T PocTexnSAaT, 1958.
42. C &Ak E?,.A! Nook, E.F., A V.,h,iJQ?L- Q* v844 L4ikaAg
FincorJ-inks, 13eft Skit. rem."-a..4 ago13S9,`333-9c7.
43. '(roeJ '. aI .+aoo$ K. C* j pgAadAoX cOwite r.h.,; ka /~caf.~t 84
? cagotin ca1o'r, 1k4 McA CokAi. T, f9S8,9i-Ht
44. kQm,n4 Ha (2.W.,6AAsL dA4 Hcd a~-.a K - cc~'tkcc iN$ .
code& , li. S~Sf. rec~.N. ~.~ 23,19S o', 111'f-1bo
45. HrJIOM A.M., firJIOm K.M. - BepofiTHOCTb H HH(0pMa1;HA,
RSA. -oe, M., FocTexxaAaT, 1960.
46. Iofo4s K .~ .0 ic. IA 8ek 0ltS4/ & :r-t fo'T ceEioN
4 kc 1. ) ZfC4ooCicbE uMc! ~a~fpbt4o~?sc ka. kCrko?e?,
~CL+h.jKn? ~ 131 SSS3, 12S-13,S
47. aosk,; A.2, 4 -,,oie oa u.~i t SOMIUC!S 4011 X& K';1U4~4
d4?stcL*ca coal.!, 711s. aueJ Cokd-t..* 1, Is Sir, 2g9_
.48. 1OwKeBHn A.A. - 0 npeAAJIbHNX TeOpeM8X,_ CB$I38HHHX c no-
HATI4eM BHTgOnHH iteneg MBppKOBa. Yen.
MaT.H8yK, , 1953, 177-180.
49. V gMa P a ?,,KPto$ .s. a j E .o p ~ o$ c1-fo'Y'we " a uct
*k& oa,4.t 8m& ph e$~4rac , ~ ' . o f .~4pp~
f k~s., 2S, t9S~c, 1`i3g-lY3g . '
50. Ke T. I.- J4 $Lw t.N ~ G~ 9 f h ~-0'SC~Mtc 4 ?tJ~,t
-'cqA ) U& S%s+. jec,It,N. ~, 3s, 19S6, 317-326 .
51. K u.?~eocJk s . 7 N ~o~t~,a.E. o~ fh~vtic~ an, aI s foc~* s he L,
Sok 4, Wi j Cu%'( &OU.C13mc.'-y, C/fAa.k,c~, *
"al Kull / LEA, Lkot, ISS3
52. KoJiMoropos A.H. - HexoTopbe padOTH nocJIeAHHX JieT s o6-
JI80TH n eAeJIbHHX TeopeM TeOpMH Bepo-
ATHOCTek. DeCTH. lily, A 10, 1953,
29-38.
b3. KojlMoropoB A.H. - Teo HA nepeAacH HH(10pM84KH- CecomA
AH CCCP no H8yytIH.IIpO6JI. BBT0M8T.npo-
eceAaHKA,
KSBQACTB8 ,.1956, I HBp666-99
M. Y1SA? Ali CCP, 1957,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80TOO246AO11700340001-4
54. KoJmoropoB A.H.
HOBHA meTpNtl@CKNA NHB8p18HT rANHamm-
tI@CKNX CNCTeM N aBTOMOpINBMOB npo-
OTp8HCTB /-eV 0
'S
M/ ~ee s - /!? t- s
as
N a B r,{ (k, Sr
COpMy)IMpOBaHHLIe Bmme ~axTt BIITeKaMT H3 o6igeia TeopeMa I, ecJIIa
ee IIp AeHHTB K BeJI KHaM
s,t _ ~tt; t
H ,000TBeTCTBeRHO,. K
j9) , 3aBHCMM OT
,
BCnoMoraTeABHOro IIapaMeTpa V.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
run UU1WWtdrs. %#-- _-
12 -
m cj*'a2HO# BeimidHOt x upeo6pasoBaBIReM CABS $r ,
? 4. ItEHTPAJILHA$I IIPEBFJILHAH TEOPEMA IH HEKOTOPUX
CTAI.DIOHAPHbIX IIPOIIECCOB.
IIycTb upogecc (f') ( IIapameTp t IIpIHHmaeT Ae ixble 8Haaee-
H8R) cTa=OHapeH B y3KOM cmHcie 1 o6AaAaeT CBOICTBOM ( 2 )?.
IIycTb CJlyiiaYiHaR BewuHa 7 A) HsMepriMa oTHoc Tedibxo ir-a.AreO
pH ir7'x 0) . p (St Cu) ecTb cTaIwoHapH} ! IIpoitecC, IIopCJt=e-
IIycTb
gyp- D/ /too
Ecim
6HCTPo9
K=~
y (3 (F) B ycnoBBM ( 2 ) y6IBaeT AoCTaTorHo
E12)
6z _ /y~2 t z /~.?re) t CIO
16parTMOBHM 6Lma ycTaxoB.neHa cazejUMU= Teopema . '
TeopeMa 5.
,
EC= BHcuo.lmeHH Tpe6OBaHBR ( 16), ( I?) ' ( 18)
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
- 13 -
$ aTOrO pe3vJlmTaTa BHTeRaeT, B gaCTHOCTXq IjeHTpaaBMH upeAeJlbHas
oj- ?7 ek
IIycTb cnygaikui a Bewraax ?C f) paBHOmepHO orpaH> eHU,
Ii,, (f) /:e c
_
(23
Y2
(a)=e .~laJ"' 't~ 2w .z e
O ?. '
x EcJJ /u/ >K;f ZOz , TO _e
upm HeKOTOpom C > o.
Teonema 6
Ecnnt Be xmEm
H rpome TOrO ,
B ~S3STePBaJ1e
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 F% a point 0(cc) of the interval
- 9, t + 0) satisfying the equality xt(
Xt t(-
Since At we have by virtue of (9) the equality
( ) P( t} = 1.
Let now T = (t j) (j = 1, 2, ...) be some denmacrsble,
dense set in I,.. Relation (;.) L-plies
(10) P( ) :.= 1.
CO
j =l
Since the process is separable and, in virtue of
satisfies the separability conditions it follows fro 3 (10)
that almost all sample functions xt(u) are jump functions wlm.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
-6-
Before formulating theorem 2 let us remember that the
function act(or) is said to have a discontinuity of the first
kind at some point t if xt(w) has at t both onesided limits
and they are not equal. If at least one onesided limit at t
does not exist the discontinuity is said to be of the second kind.
lb2an 2. Wt (xt, t e 10) be a real. s2p#Labj2 stochastic
process ag ee relation
(11) 1li'm Ob(I,e) - 0
hold for s s 0, e I G 10. Then almost all saaec-le
fmatlog-s have go dL,sgonntti Cy of the first kind at
find but arbitrary point t s I0.
!E22g. Let relation (11) hold and let the assertion
of the theorem not be true. We have then for some point
t s 10 (different from both endpoints of t) the equality
(12) P(lam zt - 0_0 0 ztO+0 - tin xt) - a
0 0
share a > 0.
Denote by J c. 10 an interval having to as its midpoint,
by An(J) as w--set for % ich 1s3(w) > 1/n, and by A(J)
on w-set for which acj(w) 0 0. The sequence (An(j)) 'a
increasing, hence
.0
A(J) - v An(J) - urn
An(J)
n=1 0--woo
(13)
P(xx 0 0) - P(A(J)) - lim P(AA(J)).
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Taking into account (12) and (13) in obtain
a (~ ..,.0 P(xJ 0) J -- 0 Ila ~ P(~in(J)
Consequently for IJI sufficiently small and n sufficiently
large, that is, for a sufficiently small, P(An(J)) will be
at least now a, which contradicts relation (11). Homes
a = 0, and theorem 2 is thus proved.
Let us remark that if (11) holds, any set of parameter
points dense in 10 satisfies the separability conditions, and
consequently for any fixed but arbitrary t a IO the equation
(14) P(xt-o = xt+o A xt) - 0
holds. Nov let (11) be satisfied and let the process (xt, t a I0)
have a fixed discontinuity point t', that is, for some 15 > 0,
the equation
(15) P(lim xt - stt) = 1
t-* t'
holds. We than obtain from theorem 2 and (14) the following.
Co, e1 Let the rsa1. wearable process (net, t a 101
A tisfv (11) NW_ is it have a discontinuit int _fjW t' a 10. ,tom pro biU s at to a aipcontinuity
of the second kind sauals 0 p i iven i (15) ?
Theorem 3 which we shall now formulate is a stronger
version of some results obtained by the author (7). It was
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-g-
assnmsd in that paper that the process under consideration
had no fined discontinuities. Now this assnoption is entirely
omitted in part (b) and is replaced is part (a) of the ran 3
by the rw bw assmptioo (11). The former assuaption is
now Obtained as a consequence (Ii). In obtaining these
atssagthset~rd results essential use will be made of the
corollas" to 4rao1 2.
3. Let the sto at process (st, t s I0) bk
(a) if = than (11) holds go say s > 0 mod if ma-reo es
(16) 1(a) -to
, _fo r ~r___vo~m~_iaterya ; 1 c I0.
(i) alssuxt fit (P) mode fUDctions of he n -a
am lu~r ![echoes,
de gamma a me --Ono -22ARM () ofd t,
(17) E4(I) - 1(I) - A(1)
Mug,_ t(1) di xt(w)
an; the 1w,1 it
am
(iv) a art gob th+ MI__ di in tip e. it
gi st(m) -t Masidad ti is rrcis has ssobsbittt9 1,
(v) the fueration q(t) existswl alsoat `vsaEre in I
.~r Ylrr__ r ~r~~ o~~w.~r+++i++
and satisfies t_,relatiot
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
(18) fI Q(t)dt 4 A(I).
(b) a(1) In Sa fik1AU j1 SMAINM ; , _ as
intasval as a( I) c ~..WM S E
aff2gLUMM (L) ,& (v) ar s A". and a or
(19) fI Q(t)dt A(I).
.qd. in the proof of assertion (i) given formerly
in (7) the ass*ption that the , c :asr has no fire! points
of discontinuity was used only fotr stating that any dererable
dense set of points t a I satisfies the separability conditions.
Now relation (11) is sufficient for this purpose and therefore
t tert.wn (i) is proved. alto r, in proving assertion (I)s
in (7) it was also proved that almost every sample function xt(w)
has a finite number of dia+aawetiaa ties. Hence by virtue of
relation (11) and the corolla*y to theorem 2, the proc.iss has
no fixed points of discontinuity, which proves assertion (ii).
Owe this fact has been established the pnf of the remaking
assertions of part (a) givea In 171 resins valid.
In order to prows pars (b) of the theorem, we remark
that if a(x) satisfies the -sehiits condition it is (see
p. 287, (2)) an absolutely ooatia o~as function of an Interval
and this Implies (see p. 287, (21) that relation (16) bolds.
Since (U) evidently holds there also, A".1 the assumptions of
part (b) are fulfilled and consequently so are all its
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
-10-
assertions. Moreover the continuity of a(I) implies (us
p. 289, [21) the absolute coutiueity of A(I), Bence relation
(19) is true. Theorem 3 is thus proved.
IMMM 4. (at, t e 1) jbt-4 Feel. ,1~~. 1e stoehaat_i_c_
Process. The relation
(20) P holds if pd only if the
tion
(21) Q(t) = 0
golds uailo 1y with resuct to t t I.
ZM". Let relation (21) hold uniformly with regard to
t s IQ. We shall show that a(I) is then an absolutely
continuous femction of an interval. Indeed, relation (21)
implies that for any a > 0 there exists such a is s 0 that
if III < a the inequality a(I) < a$II is true. Let us non
divide the interval I into a finite nunbw, say n, of non-
overlapping intervals Ik With IIk1 < a for k = 1, 2,..., no
The function a(I) than satisfies the Lipschitz condition in
every interval I,k and consequently a(I) is an absolutely
continuous function of an Interval in Ik. It than follow
that a(I) has the same property In the whole interval I.
2yvirtue of part (b) of theor&z 3 the relation (19) holds.
Taking into account (21) ve obtain A(I) = 0, which in virtue
of (1T) iaeplits relation (20). The "if" assertion is thus
proved, the "only if" assertion is evident,
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
t %!ifi4 E3 JI Wea ail. Uirv v.....,
Lot Illo I I M,c a b" to WS aw Wit In it
0
l+~lstLss s iii f (I?s) .y twt
ousts Aw am peso
t !`wart
S 1t Ma"m to id A& Ilk lime
10) in
O&A
W $ . o9 l (") IP s (.)
bm mad" at
w at S to tot : 1uM (U)
lriLr Vm(t!)>r- '1' *. (ti ( .,~~- b
list vlat%*
am"" %bo $a logo oil ft
1-soft i I" 4..m * Raw" by 3.ft "owns.
tsk, r 1 g) b- #a(t-s) ' t Mr .! ista! -s
We bwe
It soups l t 1asiwr ( l11rI a= 1 l ---so 0 wt svs
1f i - ( s) - n. as 14lli a do -mme of
s`
a1.: is
tom) (liar ..{ ; )?0)?1
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-U-
""0 Oft" pa can tabs s~wtiatss~war valawss, relation
($) lags this the &-set A(4)# tick is swh frr say
o s A(s) *Mrn a fists a J? Ab& my depmd an s asi +o such
that for j I. !a the ""lit, (T,s) - 0 holds, sfies
the r.lati* P(A(s)) ? 1. 400" s *$king Into soars ,, t
asthsd of IN uctiaa the srvsls I, swt me Oat
ala"t all (P) lop(a) brass as dissOutiumltiss of the first
Mud h a j* poem s. ~sasldaar >Irw~r a ss wrase
of positive ostwstants (* o) thaw. s*10. Tbs. set As sash that
the st(.) ?slt po.disi to 0 s A hove no dissoaeti~wdttiss
of the first kind, is gives by the fosmla
(25) A - (11 A(R,s).
Cons is*, Sao P(*(sn) a 1, IN obtain P(A) = 1. Theron
s is two p rVwd.
Lot as h erb that, as VmftWIWLa (3) has p1wed,- the
assertion et thasro 5 hors under the assuoption that as
ac--? 0 stoma relation
cP(I"t+ac ~' Nth > s) - G(At)
0
t+&ts 10
holds fiat any $ > 0. We shtli. show that relation (26)
iaplies sOUtion (22) . Indeed it tolloW from (26) that
for any w r 0 !there exists ausk aa- a > 0 that feat arW
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
? 0-
t ? lo, t + At * to with I141 is WS boar F(Ixt - %1 >
s 1i I At I . IS SAW Mrxds foot saffisiwtiy lase ? we t4r11
basfxk1#...,sthes" tiftb(zak.s) s) < q
k-i
will be emo we have
a 1k
(30) 2 F I > 6) - ac P(I I > s tst - (N)).
tai 3101 19"1 j
Let Job* - Integer such dot (1/*` ?) < s,1 sad oww"r the
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Partition (tom) of the iatsml tool] such that J LkI s i.e fit'
l
we have =m j ? 1, go-*** j
( ) 'Pti l>s st (mj))-0.
Coasegwatly
(312) k pt"! >s) s ZP(I t ssixt - st(~))*
b4 _ +1 ic?1 " k
Uses, as follows definition of the at(gI ),
the s00fled , AM* k ruts from. I to UP' an the side
of (32) am aaly be at west *got to 1/., and takig iato
sacorart the method of choosing j0, relation (32) implies
relation (29). lance relation (22) holds.
'?
t -a- Mm ltll;
we shall deal in this section with stochastic processes
of certain special types.
6* (nt, t t ZO) a..~.? 11lZ
ystie*a3ak ... ties (22) )M!' r.~, , umal1 (p )
sarla f ad s z (a) of M. at.
The assertion of As thomssa fellows from
thastrea g acid ftms thaora of loch (see chap. ? (4) )
stn g tjwlt the set of sapl+e ' Smstians zt(w) of a
separable *Wtisgale thin have s sly disasotiauitios of the
first kbWv if say, has prof abd1N ity 1.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 rri
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-15-
Dobrushin (3) has proved the assertion of theorem 6
under the stronger assumption (26).
We shall now strengthen slightly a theorem of Dynkin (5)
and Xinasy (9]. Denote
(33) C(h, s) - s _ P(Ist+nt - xt l 3- s lit
f4tl h
WOW"
,,1 P(IXt+at ` :'t I
' slit - z)dr (s),
slur. Ft(s) - P(sr - s)
IbMS 7. !dM (st, t o lp) a rMElor pgopaaa .
of h --*. 0 I?! t3wa
(35) D(h,s) - o(h)
of
~. We have
(36) P(Izt+tat - xt I * s) " I_: P(Ixt+nt - Xt I > a Iz - z)d!'t(s)
Relations (35) and (36) imply relation (26) , hence, a fortiori,,
relation (22). By a theorem of Fuchs (8) the assertion of
theorem 7 is obtained.
Ne remark that Dynkin and tinnsy have proved the assertion
of theorem 7 under the more restrictive assumption that C(h,s) - o(h).
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA- RDP80T00246AO11700340001-4
-16-
Theorem 8 below is kaoe, However, we shall give a
simple proof using tools applied in this paper.
The 8. (pct, t s IQ) bs Aral. ssto stir
AmerwoAte ad !At relation (6)
MIA -I .ash *s'a (i) (v) of t ror~ 3 t SM *
Wd? It has been proved in (6) that the independence
of increerats a*d relation (6) iaply the relation (16),
Consequently if the assumptions of thsorma 8 are satisfied, all
the assmptions of part (a) of tho 3 are satisfied and
hense all its asserttonsx- . re true.
Munn 9? lag (*t,
< < t < + 01 Thm 111=41 all ImtUms of
arcs 9 ?
Col. We have for any real t and z the relation
t38) ;(xt. , - xt)2 - 2[D2(no) - h(r) ].
Taking into accomt (37) we obtain for arbitrary t
(39)
a( - )2 - O(1s11+6).
BY the result of Kolmogor v (published in [11]) relation (39)
Implies that alwost all sample functions xt(w) of the process
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
ti
1(7 - ONLY
-17-
are sostinusus in waxy finite closed, int zv.1. Tbomm
9
is thus proved.
We ramssk d wt if the sovariaacs *motion R(i) is tw ss
differentiable at s ? 0, than almost all sa"le fssnstlsms of
the process we abselutsly eoatiaa s ( ass p. 5360141)*
We rsMWk finally that--Ss lelaysw (1) has ehsia--if d W
paressss i st, ? ?? 4 t ?e 0) is real, sapsuble, statiossry and
Gaussian and it t islation
(40) It(s) - O(xo) + 0
n
with i s 0 belch,` then almost al maple fuoctieaa are
sastia sus. Relation (40) is osdantcly leas restrictive than
relation (37) s 1erewwa islayorv o s# only Gaussian "Otte""
prsesssea.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
..19-
[93
Jot* amw, "c"tiavity p .rof aaopl. Imbi t o of
Msx processes," Tr a.I Am. Math. an., Vol. ?k
(1953), , PP. 580-,02.
(103 A. , Adw Nahum" wwww NO
$arlin,r , 1933.
11) E. SUITM, "Alclw& proposisioni aislla th oria dpi
fawsissi aL.atori* " ,ti n. ```` Ita ~??_- i
VOL 8 (1937) , pp. 183-*99.
(12) M.M. TN~h1111tsaiv, "Weak o goose of stochastic Wises
i sss taw~l. lrwativwo have wo discootinnitiaa of
the Seseod kind and the secall d 'h sristia Opts-ft-c- I
to the Ilparaestrie to$U of the Morose roar-~tt -
typo" (in #~ssiar4, ,
Vol. 1 (1996), PP. 134'1610
'17
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
run ~?a'1CL
REMARKS ON PROCESSES OF CALLS
By
Czeslaw Ryll-Nardzewski
University of Wroclaw,
Mathematical Institute,
Polish Academy of Sciences
1. Introduction
The theory of processes of calls is highly developed. In
this paper I am going to consider some questions which, to my mind,
have not yet been analyzed sufficiently from the measure theoretic
point of view.
Palm [11 dealt with a special kind of conditional probability
for stationary processes. A. J. Khinchine [2] presented and
completed the ideas of Palm. Their methods were simple and elegant
but they were of analytical character. In this paper I am going
to give a different, and so far as I know, new approach to these
problems. I shall confine myself to considering some basic notions
and their properties.
As a byproduct I have obtained a result from ergodic theory
which seems to be of some interest in itself.
2. Discrete time
From the measure theoretic point of view, the theory of
stationary processes of calls with discrete time is quite simple
and consequently it is not dangerous to omit some technical details.
Let us consider a doubly infinite stationary sequence of random
variables taking only the value zero
or one. It is easy to prove that either there are no calls or
FOR
'n P f ",1V
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
F ~"n raJr~a~. ~dd.n LY
2 -
they occur infinitely many times in both time directions. In
symbols
(i) P(>; 0 or lim s; = 13m . = 1) = 1,
i i>-+oo iT-oo 1
where P denotes the probability measure. The first possibility
is uninteresting from any point of view. Hence we may suppose that
(2) P(~i = 01 = 0.
Further, the general case can be reduced to this case by the
introduction of a "new" probability measure, invariant under the
shift transformation,
(3)
P'(,) df P(-IN), where N = (ei = 0).
We denote by A the event 0 = 1) and put a = P(N).
Under the condition that A has occurred, that is, that there is
a call at time t = 0, we can define the sequence of random
variables ???,~l_2,~_l,T)A,T)l.,???, which are equal to the distances
of the successive calls. The enumeration begins from the call0.
This is illustrated in Figure 1.
Figure 1
~w l:C~ tm
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 3 -
We denote by p(?) the conditional probability
p(?) = P(?1>;0 = 1) = P(?IA). We can now assert
Theorem 1. (i) The sequence (ni) is
(a) stationary with respect to measure p,
(b) the random variables ')i admit only positive,
integer values,
(c) the expectation of ni is finite (and is equal to
(P(~0 = 1))-1).
(ii) The correspondence between (ei) and (qi), or more
precisely between the probability measures P and p,
is 1 to 1.
(iii) Each sequence (ni) of random variables satisfying (a)
to (c) can be obtained in this way.
Proof. Let S_m,S_m+l,???,S0,S1,???,Sn be arbitrary positive
numbers. In order to prove the equality
(4) P(71-m = S_m,...) r1n = Sm) = P(rl_m+l = S_m,...,%n+l = S
it is sufficient to apply the shift transformation S_1 times in
the formula
(5) P(sm = 1'~s+m = 0,...,~s-m+1 1'...,s-1
~0 = 1, it=o, s1=1,...'~s =1).
n
Hence property (a) has been proved. Property (b) is obvious.
Further, by the stationarity of (fin) and formulas (1) and (2)
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
FAR
_4-
00
00 (6) Ep(710) _ p{~0 -la 0 = 1, =0,..., i-1-0}
i-1 i-1
1 CO =
a z P(~_i1=0, 0=0} a.
i=0
Thus (e) is proved, which completes the proof of the first part of
our assertion.
Now we shall consider the correspondence between the
probability measures P and p. We have
(7)
00 AO
p(e) = E P(ID-i-1' -i+1=0,.0.=O, e)
i=0
00
= =0, ..., P(i)}E P~> 0=1,~1 0,
i=0
where e(i) denotes the event 8, shifted to the right i times.
Finally, we obtain from (7) the following formula expressing the
probability measure P by the probability measure p
(8)
00
P(s) = a Z p~710 >
i=0
Now we have to prove only the last part of theorem 1. Namely,
we suppose that conditions (a) to (c) hold, and formula (8) can be
regarded as a definition of the measure P. Evidently we must put
a df (f 110 dp)-1. We shall compute the value of P(e(1)). We have
P(g(1) 00
(9) ) = a Z p{i0 > i, ~(i+1)}
i=0
00
= a y- lpko, i, F,(i)) + p{i0=i, ~(1)} -ap(P-)
i=0
00
= P(e) + a E p{r10=i,
i=0
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FOR U~
- 5 -
The last brackets, in virtue of the stationarity of the sequence
(r1,n) with respect to the measure p, are equal to zero. Hence
we get P(e(1)) = P(g) for all events P-, that is, the measure P
is indeed invariant.
At the end of this section we give the law for forming
statistical mixtures of the measures P and p.
Theorem 2. If a measure pi corresponds, in the preceding
sense, to a measure Pi, then the measure
(10)
(Z ai~i) -l E aj?,iPi
i
corresponds to the measure Z-A Pi. where Ti > 0 and Z.? = 1.
The proof of this theorem is not difficult. This rule has,
however, an important consequence which is not quite evident.
Theorem 3. The sequence (ei) is metrically transitive with
respect to the measure P if and only if the sequence (71i) has
the same property with respect to p.
By. metrically transitive we mean that each event concerning
the variables ~i which is invariant, under the transformation
(~i) (~i+l), has P--probability equal to zero or one.
Proof. The set of P-measures and the set of p-measures are
convex. From theorem 2 it follows that the extremal points of one
of the convex sets are mapped onto the extremal points of the other.
On the other hand, extremality and transitivity are equivalent,
which concludes the proof.
This elegant method, based on notions and theorems of the
theory of convex sets, was successfully used in ergodic theory by
Savage and Hewitt.
MI
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
-10
L ~,::;a~
6 _
The recurrence transformation
We shall consider a probability space (2, B, ?) with a
measure preserving transformation T, that is, a point
transformation from the space S@ into itself, satisfying the
following conditions: T-1(B) C B and ?(T-1(e)) = u.(e) for all'
. E B. We shall say that T is 1 to 1 measure preserving if it
is 1 to 1 from the space 2 onto itself, if T(B) = B and if T
is measure preserving. Let E be a fixed measurable set of
positive measure. By the famous Pbincare-Zermelo theorem for
almost every point w E 9 some of its iteration Tk(w) for
k ? 1 also belongs to the set e. More precisely, we have the
relation
(11) limn E X~,(Tkw) > 0 for a.e. w c E,
n k=1
where X8 denotes the characteristic function of the set e.
Hence we can define on the set 8 the recurrence transformation
TP_ by condition
(12) T8(w)d=f Tk(w) ? 8 and T1(w) j e for 1 S i < k.
More exactly, this transformation Te must be considered on
the set
(13) 8{w: w E 9 and lim X (Tkw) 0 = 1}.
k
Now it is easy to verify that F, and 90 are almost equal. In
our probabilistic considerations, sets of measure zero may be
neglected. We emphasize that the recurrence transformation depends
on the choice of the set a and is defined only on it.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09 :ppCIA-RDPg80pT00246A011700340001-4
ir RPn`lrf~ 1
;t
- 7 -
The basic property of the recurrence transformation is
Theorem 4. T. preserves the measure ? in the new measure
space (e, e n B, ?).
In probabilistic applications we can also consider the normed
conditional measure ?~(?) 41 ?(.? ). For the proof it suffices
to verify that
(14) w( ) _ o Z o ?[e, n T-1(F-c) n ... n T-k+1(~) n T~k( )] .
k=1
It is easy to see that the right side of formula (14) is equal to
[T-l()1?
On the other hand,we obtain by a simple computation
(15)
n
Sn = E 4[[n T-1(ec)O ... 1) T-k+1(ec) fl T-k( ),
k=l
= ? [F nn T-n(36 )1,
where Fn = U??? UT n(g). We observe now that the limit set
limn Fn = F = U on?_O T-n(e) is almost T-invariant and contains the
set . Hence we can write
(16) Sn = T-n(,gk )1 - ? [(F-Fn) /1T-n(,
= ?(. ) - 41(F-Fn)n T-n(,X)I.
To conclude the proof it suffices to remark that the second term
on the right side converges to zero.
It is easy to observe close connection of the recurrence
transformations with the theory of stationary sequences of calls.
For this purpose it is enough to consider the exponent k, in the
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
er-
$ ,L ONLY
r
FOR 0
definition (5), as a function of a point w E 80. The measurability
of this function k = k(w) is clear. Now we form a sequence
e),...,
(17) k(w),k(Te),k(T2
In view of theorem 4 this sequence is stationary with respect to
the measure ?., and has exactly the same meaning as the random
variables r0,n1,r2, " ' considered above. We must put 94(e.=1),
Td=f the shift transformation and we have p df P and ?e = p.
In addition we remark that for each 1 to 1 measure preserving
transformation T the transformation T. is also 1 to 1, and
therefore we can also form the negative iterations
(18) ...,k(T-3w),k(T-2w),k(T-l(U).
We are not going to give a systematic study of the recurrence
transformations. We shall present some formulas and properties
only.
(i) Te 1 = (T 2 )8 1 for measurable sets F-1(=g2?
(ii) If the transformation T is metrically transitive then
T9 is also transitive.
(iii) If we suppose in addition that the iterations of 9
cover the whole space S@, then the inverse implication
is also true.
The proof of (i) consists of a simple calculation based on the
definition (12). The proof of (ii) and (iii) is the same as that
of theorem 3.
We can raise different problems about the recurrence
transformations. Examples are various mixing properties hereditary
from T on T.
. ~, cam, ^' ' r'? !, ~ flc [lE3
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
9 -
4. The c nditional robabt for arbitrar rocesses of calls
We start with a precise description of the measurable space
for the process of calls. Let it denote the class of all
countable subsets of the real axis R, which is the time axis,
that are finite in each finite interval. The elements of Q,
which are the realizations of our process, are denoted by w.
By N(w, Q), or simply by N(Q) we denote the number of calls in
the time set Q, that is,
(19)
N(w, Q)af card (w 11Q).
Now we define the class. B of measurable subsets of Q as
the a-field generated by all the functions
N(., Q), where Q is
a Borel set, that is, B is the smallest a-field with respect to
which all functions N(?, Q) are measurable. It is obvious that
in the preceding definition we can replace the family {Qj of all
Borel sets by the family of all intervals, or by the family of the
intervals with rational endpoints. Evidently each w can also be
treated as a purely atomic measure, finite for bounded sets. We
emphasize that (2, B) has good set theoretic structure. Namely
it is not difficult to prove that a can be mapped by a 1 to 1
function onto the unit interval I and the class B onto the
class of all I Borel subsets. Hence the typical difficulties of
the theory of conditional probability do not occur in our space
(2, B).
Let a fixed probability measure P be defined on (0, B).
Our next aim is to give a precise meaning to the notion: the
probability of an event 8 under the condition that a call
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
~=. ~L
FOR P"n--t
- 10 -
occurred at moment t. For this notion, not yet defined, we shall
use the symbols P(?4t) or Pelt E w). Now, we introduce a new
assumption, which is quite natural and at the same time seems to be
necessary for our consideration. We suppose that
(20) fN(uQ)P(dw) = EprN(Q)J < co
for all bounded sets Q. Consequently we put
4(Q)df
EPON(Q)I.
Obviously ? is a Borel measure on the time axis.
For each event 8 E B the integral
(21) f Xe(w)N(w,Q)P(dw),
treated as a function of the set Q, is an absolutely continuous
measure with respect to ?. Hence by the Radon-Nikodym theorem
we can write
(22) Jxe((c0Q)P() = fQ P(P-I t)p(dt) .
For each fixed E the Radon-Nikodym derivative P(81t) is unique
only a.e. with respect to p, and we can always suppose that it
is a "true" measure with respect to sets . E B. This follows from
the previously mentioned property that a measurable space (a, B)
is a Borel space.
Formula (20) can be generalized to
(23) ffNQ)Pdw) = f ?(dt) f f(w)P(dw)
Q
where f is a P-integrable function.
It seems that this way of introducing the probability P(elt)
is appropriate. We shall only remark that
n "c a
% 1j. Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 .~ ~f
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 11 -
(i) If there is exactly one call then P(et) -is identical
with the ordinary conditional probability.
(ii) If some process of calls is realized by the sequence
of random variables x1,x2,???,
for which Prixi # xji
= 1 for
i # j, in the following sense N(Q)4f zi yy(xi) for all Borel
sets Q, then our assumption (20) takes on the form Eil-ti(Q)
= ?(Q) .e co for all bounded Q, where lai(Q)df Pr{x;l E Q3.
Moreover, the probability P(elt) can be written
(24) Wit) = E P(EIxi = t)P(xi = tit),
i
where P(xi = tit) is equal to the Radon-Nikodym derivative
d?i/d? and can be interpreted as the conditional probability that
the ith call occurs at the moment t, given that a call appears
at this moment.
The stationary Process
Now we shall consider stationary processes of calls. We
shall use the following notations for shifts
wt d f
w+ t for w e t and - o ,-' t< oo ,
(25)
w t df -t
E E E.
We add the new assumption P(P) = P(9t) for all 8 E B and
-co < t < Co. As in the case of the discrete time, we have
(26) PtN(-oo, +oo) = 0 or N(-oo, 0) = N(0, co) = ooj = 1,
and in what follows we always assume that
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
to'
",b~r ~~i~ Ahtf
12 -
(27) P(N(-oo, +oo) = 0) = 0.
Hence we can restrict our attention to the realizations cu with
infinitely many calls in both time directions. First we shall
establish the form of the conditional probability P(elt), from
the preceding section.
Theorem 5. There exists one and only one probnbility measure
PO defined on the space (91, B) for which the measure function
(28)
TT(EIt)af PO(e
depending on the parameter t, satisfies the condition (22) for
all 8 e B and Q.
Proof. For the stationary process the measure ? is
proportional to the Lebesgue measure p(dt) = adt, where a is
the intensity of calls. Roughly speaking, the matter is quite
simple: the new measure PO(e) is equal to P(eIO) and formula
(28) is a special case of
(29) P(elt) = P(FsIt+s),
which seems to be obvious in view of the assumed stationarity.
For a precise proof, let P(elt) be any conditional probability
measure satisfying (22). It follows from the stationarity that for
each e e B, s e R and for almost every t, in the sense of the
Lebesgue measure, the relation (29) holds. In view of the Fubini
theorem we know that there is a number t 0 such that, for almost
every s and for all 9 e B we have P(81t0) = P(85It0 + s).
41 'A' U -
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Nil
L t6
_ 13 _
The quantifier "for all 8" can be put at the end because P(?It)
is a measure and it suffices to consider only some countable class
of sets 9 generating the whole field B. It is easy to see that
the measure P0(8)dfP(et0It0) satisfies the assertion of theorem 5.
The proof of the uniqueness of the measure PO is omitted because
it is quite simple.
Remark 1. Now our conditional probability measure P(?It)
can be determined in a unique manner by equation (29). This
"regular" P(?It) will be used later.
Remark 2. Now the equation (23) takes the form
(30) ff()N(wQ)P(dw) = of P0(dw)f f(wt)dt.
Q
Next we give another description of P0. We say that a B-measurable
function f(w) defined on Q is continuous if and only if it is
bounded and if for each fixed w e Sit the function f(wt) of the
teal variable t is continuous.
Theorem 6. If f is continuous, then
E I,
Jim I f()P(dw) = ff(w)PO(dw)
III-a-0 a I N(I),1=1)
an interval. Or, in another form,
lim EP(fIN(I) A 1) = EPO(f).
III-4-0
Proof. From formula (30) we have
(33) lira a I
III -_-,_0 f f(w)N(w,I)P(dw)
0(dw) lim - f f(wt)dt =f PO(dw)f(w),
III---o
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
and on the other hand
- 14 -
a I
(34) Iaif 1[N(I))
< sup If(w)I a I f [N(w,I)-11P(dw).
W062 (N(I)al)
The right side tends to zero together with the length of I
(compare the theorem of Koroliuk, p. 39 of [21).
Corollary. PO(20) = 1, where nodf(w: 0 E W).
For the proof of the preceding very intuitive equality, we
consider a sequence (fn(w)) of continuous functions defined as
(35) fn(w)
f 0
1-nd(w) if d(w)
if d(w) A
`n'
where d(w) is the distance of the set w from the point t=0.
By theorem 6 we have ffn(w)PO(dw) = 1 for n=1,2,---, and hence
in the limit we obtain PO(20) = 1.
We introduce now a sequence (%(w))
the subspace 00 sketched in Figure 2.
X10 111
Figure 2
of functions defined on
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
V~a
- 15 -
7
Th
The random variables
) n=0,?1,?2,--- in the
(Ti
.
eorem
n
measure space
n B, P0) are
(53
, Q
0
O
(i)
> 0,
nonnegative 'fi
(ii)
n
they have finite first moment
EP (rin) = a-1 < co,
0
(iii)
''' form a stationary sequence with respect to
n
'
1
2
the measure P0.
Conversely each sequence of random variables fr1n1 satisfying (i)
to (iii) can be obtained in this way. Moreover, the correspondence
between P and PO is 1 to 1 and it is given by
X10
(36) ff(w)P(dw) = of PO(dw)f f(w-t)dt,
%, which is valid for all measurable and '-integrable function f.
Proof. Let a be a positive number. We have, in virtue of
the stationarity,
n
(37) f(w)P() _ f f(w)P(dw)
{N -a,O)>O) k=1 (N(I-k)>O,N(I-k+l)=O,...,N(I-1)=0)
b E f f(w-(k-l)a)P(dw),
k=I (N(I-1)>O,N(IO)=O,...,N(Ik-2)=O)
where Sf a/n and Ikdf [k8,(k+l)b)I. Consequently, we obtain
mn(w)
(38) f f(w)P(dw) = bf P(dw)S E f(u~-(k-1)a
{N(-a, O) 0) 0*11
where mnd-f n [cp/n] - 1 and cp df min(rIO, a). When n ->-oo, the
m
(39) b En f [w-(k-1)11,. ~f(w-t)dt
O
k-1
6d aya -f]] L USE i~!L
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
rr ~Q jy1 y ' ky4~, 6p ~ qpA
K T TN ~NU ~~Y YIY.Y
- 16 -
in a bounded manner with respect to the variable w. In virtue of
theorem 6 we obtain
f (N( min(r 0,a)
(40) -a,0),0) f(w)P(dw) = a f PO(dw) f 0 f(w t)dt.
Finally, if a -~++oo in the last formula, we obtain the equality
(36) for each continuous f and, consequently, for each
P-integrable function.
Now from formula (36) we obtain, by putting f = 1,
(41) 1 = a " nOPO(dw) and EP (no) = a-l.
v 0
0
Next we must prove property (iii). Let X6(w) be the
characteristic function of the event (w: N(w,I) ? 1), where
I = < 0, b > 0.
From (31) and (36) we have for each continuous f
(42) ffo)P0(d) = lim aff(w)Xb(w)P(dw)
b--~-0
lim f PO(dw)S f 110 f(w-t)Xo(w-t)dt
0
6--,,.o Q0
I _'1
f(w-t)dt
= urn PO(d
b--*0 QO
= f f(w-no)F0(
.0
x(n0-b,0)
Hence we have obtained the important equality
(43) f(w)PO(dw) = f
o aO f(Wno)PO(dw)
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
valid for all continuous f, and consequently for all bounded
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
rum
- 17 -
measurable functions since, by iterated passage to the limit, we
obtain all measurable functions from continuous functions. The
last formula expresses the stationarity of hnI Hence statements
(i) to (iii) are proved.
We shall now give the proof of the inverse implication. We
suppose that (i) to (iii) hold and a probability measure PO
satisfies (43). We define the measure P by formula (36).
710 (44) ff()P(d) df fPO(dff(t) dt
where
(45)
From (43) we obtain
TlO+...' k
(46) ff(w)P(dw) = 0 foo PO(dw)ff(w-t)dt
~0+" * pk-1
for k=1,2,??? and consequently,
(47) ff(w)P(dw) = p 1im f"' PO(dw)k+1 fO
k--co f(w-t)dt,
0
and by analogy for arbitrary real s
(!8) ff(w5)P(dw) _ P lim fQ10 P0(dw)k
k---oD
H
ence
(49) ff(ws)P(d) = ff(w)P(dw),
that is, the measure P is invariant.
T1O+? ? .+%+s
f(w-t)dt.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
FRL
- 18 -
In addition we must prove that the conditional probability
induced by P and denoted for the moment by P is identical with
P0. We have by (36)
(50) f(w)P(dw) = rp*(dw)f0
f(wt)dtf
Let Xg have the same meaning as before. We obtain
(51) fX~()P(do) = 0 f PO(dw)min(8, r1O)'
0
and
(52) lim
5-11.0
fX,(w)P(dw) = a,
limn a P fP0(c1)min(orio) _ P.
b 0
Therefore, a = R.
From (44) we have, by the previous reasoning, the equality
(53) fP*(dw)f(w) = fP0(dw)f(cu-710)
which is valid for all continuous f. Then P = PO.
Finally we give an analogy of theorem 3 which can be proved
in a similar way.
Theorem 8. The measure P is metrically transitive if and
only if the measure P 0 is metrically transitive.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
&
TOR
- 19 -
[1]
REFERENCES
C. PALM, "Intensit(tsschwankungen im Fernsprechverkehr,
Ericsson Technics, Vol. 44 (19+3), pp. 1-89.
[2] A. J. KHINCHINE, Mathematical Methods of Servicing Problems,
Moscow, 1955.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Statistical esti tion of semantic provability.
Antonin Spadek.
Let us point out that there is nothing unexpected in
this paper. The sole element of novelty is the formal descrip-
tion of a simple relation between a chapter of mathematical
logic and mathematical statistics. The word semantic occurr-
ing in the title indicates that, roughly speaking, provability
or non-provability is to be estimated on the basis of.truth
and falsehood in interpretations in models. The logical for-
malism used in this paper is monadic logic introduced by P.R.
Halmos in C2?. In principle it is possible to replace the mo-
nadic logic by a more developed formalism, for instance, by
polyadic logic r 3l. The elements, the provability or non-pro-
vability of which is to be estimated,.as well as the interpre-
tations are chosen at random by appropriate chance mechanisms,
hence, the whole problem is probabilistic in nature. The estim-
ation procedures established in this paper possess a natural op-
timum property. The study of behaveor of these procedures at in-
finity shows that the statistical decis_on functions of finite
size, which estimate provability are, in fact, asymptotically
good approximate proofs. One may hope that the questions treat-
ed in this paper reflect at least the most elementary features
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
of heuristic reasonirpg which is so perfectly realized by the
human brain.
All that is necessary for an easy understanding is de-
veloped in the.paper in full details and with. intuitive
justification. The main reason is that one cannot expect that,
in general, specialists.. in:mothematical logic are familiar with
concepts, methods and results of statistical decision theory
and.that statisticians are familiar with formalisms of mathema
tical logic.
The basic concepts and results of statistical decision
theory on an appropriate level of generality are summarized
in ?1. These results are. then applied in ?2 to the problem of
statistical estimation of belonging relations. The passage from
the considerations of ?2 to the solution of our main problem
if statistical estimation of provability is. completely trans-
parent and forms the contents of ?3.
The present paper, which is closely connected with [71.'
does not furnish more.than may be intuitively expected and,
therefore, its practical value is very limited. Further deve-
lopments in this direction, however,,will probably throw some
light into the mechanism of human behaveor in problem solving.
?1. The Neyman - Pearson. theorem.
A wide variety of problems of mathematical statistics
can be reduced to a simple application of a classical theorem
due to J.Neyman and E.S.Pearson t51. It is not surprising that....
this.famous theorem plays a decisive role in our considerations.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
U LY
u Er Yj
Its original version,. however, does not fulfil our require-
ments. The main reason is that it does not allow the discussion
of cases in which more general sample spaces occur. We shall
see later that an adequate generalization of the Neyman-Pearson
theorem can be easily obtained.
Our basic probability space will be denoted, as usual,
by ( 11) a, ew), where Si is the set of elementary events,
G the sigma-algebra of random events and the probability
measure on . The Symbol co ,will always mean an element of
. Throughout this paper the notation just introduced will
be preserved.
A statistical decision problem is defined to be a pair
(f ) of random variables, where c0 takes its values in
the parameter space and ranges over the sample space. The
parameter space is assumed to consist of exactly two elements,
namely 0 and I , hence, the measurability of is assured
by the requirement that
On the other hand, no restriction will be imposed on the range
X of except that it is supplied with a fixed sigma-algebra
,36 of subsets of X . The measurable space (X; i) is said to
be the sample space. The transformation of Si into X will
be called random sample if
{w; (w)EE}EC
for every set / from )r'.
Roughly speaking, a statistical decision is an action
determined by the value of the random sample. This action can be
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
'NIL I
FOR
formally. described using the concept of decision function.
The domain of '..a decision function is;the sample space and its
range is usually called the space. ofdecisions. In our case,
however, the space of decisions is assumed to coincide with
the parameter space, hence, a decision function d is a func-
tion defined on X and taking the values 0 or 1 . But this
,is not enough. In order to ensure that the compound transformation
orf f (.)) becomes a .random, variable, it is reasonable, to
impose on O
an. additional condition of measurability, namely,
[x x) = 1 } E'
A natural manner of how to evaluate statistical decisions
with respect to the random occurence of parameters is the con-
vention that
(~ w : c~Cw) =1 } n f w: aC Cw)) = o}) u({w cp(w)=0}n{w:~( (w))=1})
means the randomevent' of incorrect decisions.
Our. main cuestion ishow.to choose. the decision function
CJ in order to make.the probability of therendom event of
.incorrect decisions as small as possible. The answer is quite
satisfactory.
THEOREM 1. There'always exists. a. statistical decision
function which minimizes the probnbility of. the random event
of incorrect decisions.
The proof is a simple ; tpp] ic: ;tion of the Hahn decomposi-
tion theorem 4 . Let us write
v(E) ({w:a(cv)=I} n I(E))97(W)=0}n 1(E))
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T002246AO11700340001-4
MAI
for every ( from X . Clearly, y is a signed measure on
. It is well known-that there exists a set H from 3E
such that 1>04 n E) > 0 and ) (H() E) 0 for every
from 9, where H X- H. Since
(E) =v(HnE)+)) (H E) (f-1)4-V(14~_E) < v(H)
for every E from ,E , hence, the number ))(H) is the maxi-
mum of V on 3C . Now let us define the decision function h
by the requirement that
{x: 3(.V. )=1} = H.
Since for every decision function O the'probability of the
random event of incorrect decisions is equal to
hence, using the fact that (3 is determined by the Hahn de-
composition (H) H') of V , we can write
Cu,{w:cp(w)=1}-v1 :j'3()=1 ~. w:~(w)-1 -y x:~S'lx)=~
for every decision function c , q.e.d.
The decision function 3 , the existence of which is
assured by Theorem 1, is said to be the Bayes solution of the
statistical decision problem (tp) ) .
It is easy to verify that the signed measure V is abso-
lutely continuous with respect to the probability measure Ltt,f
in 6 , hence, using the Radon-Nikodym theorem 141, we can
state that there exists a real valued measurable function It
on X such that
V (E) = J , (x) oC(u.
U~,J~ IIJILJu,L? ~ f ;..~Ir~. l.,l~llLkll
u~l:tluLL, li
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
~~ a FI A-LJ LLU L'l.lyl,,-~ f~{y~
for every set E from . We see at once that the set
and its complement determine a Hahn decomposition of 7) and
this is in fact the content of the Neyman-Pearson theorem. It
is, however, more appropriate to formulate this theorem in
terms of the measurable functions 1, and , defined for every
element X of X and every set E from 36 by the equations
~w ({ w : c~ (cv) -= ~ } n -r(E)) = oG 444f -,
- S C E ) (1- ac,) .f (x) cC~t c '
~ E J
where
ci cP (w) =1
The number o(, is said to be the a priori probability in the
parameter space. Clearly, if oc >O then t is a conditional
probability density and if oC.< 1 then, is a conditional
probability density. Since
t-'{ x : & (x) = 0C~'(.x) - (i -ac), (.z) =1
the Neyman-Pearson theorem can be formulated as follows:
THEOREM 2. The statistical decision function (3 deter-
mined by the relation
{x: f3 (() = 4 } _ {-q.: o(. V(x) > (0-o .) (x)i
minimizes the probability of the random event of incorrect
decisions.
In applications of this theorem the densities and
-~ are always assumed to be known, hence, the Bayes solution
of the statistical decision problem ('f) ~ ) depends only
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
on the a priori probability oG in the parameter space.
Now we shall introduce the abstract substitute of the
concept of sample size. The classical, model shows that one
of the most important consequences of the reduction of sample
size is a restriction imposed on the measurability of the de-
cision functions. This fact motivates our definition of the
size of a decision function.
Let 96,
363).... be a non-decreasing sequence
of sigma-algebras. of. subsets of , and suppose that the union
is a base of the sigma-algebra . This sequence will serve
as a scale of the sizes of decision functions.
The decision function S is said to be of size "Vt, if it
is measurable with respect to the sigma-algebra , i.e.
{ : ( ) 1 }
RO=1}EX,y,,,
but it is not measurable with respect to the sigma-algebra ,9E
for tPYt. 4 . We shall say that Y is of finite
size if there exists a positive integer 'vi' such that T is of
size /M . The decision function 6 , which is by definition
measurable with respect to the whole sigma-algebra .6 , is said
to be of infinite size if it is not of finite size. Clearly, if
there exists a decision function of finite size then, roughly
speaking, the scale 36,, X2 , 363 must have effectively
an infinite number of divisions.
Denoting by L, the set of all decision functions in X
and by a,YL that of all decision functions in X at most of
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
8
size 7 t , for /YL = I, Z?, 3, . , , we see at once that
4cp2C' A34:: cp
hence, if is the probability of the random event of incor-
rect decisions associated with the Bayes decision function
fromQ and 81that associated with the Bayes decision
function (3, from D,y,L for M -41j 2) ) .., then
i.e., as may be intuitively expected, the least probabilities
of making incorrect decisions do not increase whenever the
sizes of the decision functions admitted to the concurrence
increase to infinity.
By Theorem 2 a Bayes decision function of size /Yt.
is determined by the relation
~ ~(: ~3,n(x) =1 } {.x: o(,.,i(.X) > (?-~) c (X).}
for /'L = 4) 3,... , where to and are defined using the
sigma-algebra % in the same. way as IC+ and 4, were de-
fined using the whole sigma-algebra
The main effect of increasing the sample size can be ex-
pressed as follows:
THEOREM 3. The sequence of random variables
3( converges to the random variable (f
with probability one, the sequence of random variables
3 converges to the random variable
with probability one, and the sequence E... Ez) F_ 3.) ... of pro-
babilities of the random event of incorrect decisions, associated
successively with the Bayes decision functions (3, (2. t33),
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
converges to the probability E of the random event of incor-
rect decisions associated with the Bayes decision function
The first two assertions of Theorem 3 are immediate
consequences of a well known martingale theorem C12 and the
last assertion is contained in 163 as a particular case.
Let us note that if. E = 0 then the last assertion of
Theorem 3 expresses the well known property of consistency of
the Bayes decision functions 134, 132 (33,???
?2'. Statistical estimation of.belonging relations.
A wide variety of questions concerning statistical esti-
mation of provability possesses a common statistical structure
of very elementary nature and this fact enables us to treat the
basic statistical problem separately and independently of any
consideration belonging purely to the domain of mathematical
logic. After establishing the general results it remains only
to interpret them appropriately in order to obtain the desired
final answer to various questions of statistical estimation of
provability. The realization of this last. step is, however,
rather only a routine matter.
Suppose that one wants to decide whether an element chos-
en at random by an appropriate chance mechanism from a fixed
set A belongs or does not belong to a fixed non-empty proper
subset M of A
The random variable oZ taking values in A is assumed
to be a formal substitute,of our basic chance mechanism. One
of the most natural requirements concerning measurability is
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
10
The direct observation on -M is replaced by observations on
the subsets Q (iYYt.) of A for /YYt. = 4 her}ce, it is
also natural to ..impose on oz an additional condition,namely,
~w= ~Cw)E QCm) ECs
for mt. = 4) z, 3) and this completes the definition of the
random variable
Now let Z been ordinary random variable taking on va-
lues of positive integers. The compound transformation
is a random variable in the sense that
(1)
fw:'~,EQ(z(w))I E @
for every element of A . This follows from the obvious
identity
where 'vn, is the -th subscript for which ti E Q (rn~) ,.
Clearly,
Oz (w E Q(r(w)) = U~(fw: ~j(w) E Q(m)}nznn ),
hence, we can state that
(2) w:~Cw)cQ(r(w>) c C~
and this is the most important fact concerning the relation
between the two kinds of random variables.
In elementary set theory the relation L. E M is. often
expressed in terms of the characteristic function C. of M by
the equivalent statement that C ()tL) A. little more com-
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
plicated concept is that of the characteristic function of a
random set. If for each /M = 1) Z.) 3').... C .(m) denotes the
characteristic function of the set Q (m) then by (1) the
compound function C(z (. )) is an ordinary random variable tak-
ing the values I or 0 . The random variable C., (-C (.)) is
said to be the characteristic function of the random set
Q (Z (. )) , The element It, of A belongs to &(v(')) or
to its complement A p (-r(w)) according as G (Z(c~))(~.) =1
or G(z(w))(~i,) = 0.
Clearly, the compound transformation G (n2 (. )) is an
ordinary random variable taking the values I or 0 . The
value of at CO belongs to /"j or to its complement A- M
according as c, (~ (w1) 7 or C, ) = 0 . By (2) the
compound transformation C (z-(?))(", (. )) is an ordinary random
variable taking the values 4 or.0 . The value of /~Z at w
belongs to Q (Z(w)) or to its complement A - 6 (r(:~)) accord-
ing as C (z (w)) (~ (& )) = 1 or C (z (w )) (~ (w)) = 0. We have
thus defined a probabilistic extension of belonging relations.
In order to simplify the notation we shall write (P.(. )
instead of C. and (.~ instead of
Let X be the set of all sequences
every term of which is either equal to ' or to 0 . The coin-
cidence in the first 'Yt, terms of sequences from X is an equi
valence relation in X . The classes of all unions of equi-
valence sets induced by this equivalence relation is a complete
algebra of subsets of X for every n The sets from
are called n/4-dimensional cylinders. Our basic sigma-
algebra % of subsets of X is that induc 4, by the union
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
4 aL W .,y
Let t1) 72)Z3,...
U36
be a sequence of integral-valued
random variables. Then the sequence
where
xc(.) = c (v (. ))(j (. ))
for /YL 2, 3 is a
) ...j
12
random vector. with. values in X . Clearly, 3& is the smallest
sigma-algebra of subsets of X for which the vector is'
measurable.
Now the ground is prepared to put the traditional machi
nery*of statistical decision functions into action. The passage
from the general scheme of statistical decision to our particu-
lar. case is very simple because the notation of ?1 is preserved.
As has already been pointed out in ?1, the Bayes solution of a
statistical decision problem depends on the a priori probabili=
ty in the. parameter space. We shall see, however, that, as com
pared with the general case, our particular version of the sta-
tistical decision problem is, roughly speaking, less sensitive
to the exact knowledge of the a priori probability, provided
that a very simple and natural condition, namely,
(3) MC a ("11)
for 4, 2, 3J...* is satisfied. We shall see that under this
condition, either the decision function (3 . which associates
with every sample point X of X the decision 0 , or the
decision function which associates with the sample point ,
of X the decision I or 0 according as the first i.. coor-
dinates of %X are-equal to 1 or at least one of these coor-
dinates-is equal to 0 , can occur..More precisely:
FOR L
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
f OR
13
THEOREM 4. Under (3) the Bayes solution of size w of
the statistical decision problem (Lp ) is determined by the
decision function or and the probability of the ran-
dom event of incorrect decisions is equal to
(4)
or to o(, according as
(5)
xm~~~ xmtz~ .... < oC, /(1-oL)
or the opposite inequality holds.
The details. of the proof can be omitted because Theorem 4
is nothing else but a particular version of Theorem 2. It suf-
fices to note that, as compared with Theorem 2, the main sim-
plification arises from (3) and from the definition of X , x
and . Under these conditions or 0 , according
as the first 7'b coordinates of i( are equal to -1 or one at.
least of these coordinates is equal to 0 , and 0 ~i,,n(x)
for every .X from x , hence, Theorem 2 is immediately applicable.
In order to make the intuitive content of the theorem
just established more transparent we shall give the informal
description of an experimental procedure of how to estimate
that an element of A chosen by belongs to M or to its
complement A- M using the.Bayes decision procedure of size /Y
i . e. that determined by the random variables t,.)
Whenever the inequality (5) does.not'hold then the value of oz
is always estimated to belong to A- M . If (5) holds then
the.decision procedure runs as follows: At the first step we
choose the set Q (IM) determined by the value of Z, .
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
o ft
GL
Y, 14'
If the value of *1 does not belong .to this set, the procedure
is stopped and the value of 42 is estimated to belong to A-Al.
In the opposite case we continue the inspection choosing the
set a (M) determined by the value of Z :z . If the value of
nZ does not belong to this set, the procedure is stopped and
the value of is estimated to belong to A - M . In the
opposite case we continue the inspection choosing the set 06n.).
determined by the value of Z3 and so on. Exhausting all the,
sets (SZ('YY.,) determined successively by the values of Z,.)
?..,r,,n, without reaching the decision that the value of 4 be-
longs to A - M we accept the decision that the value of nZ
belongs to M . We see that the final decision that the value
of 1 belongs to A - M can be reached at every step of the
decision procedure. On the other hand, the opposite decision
that the value of nZ belongs to M can be reached only at
the last. step.
(6)
(7 )
Now we shall show that under the two additional conditions
(1 a C.> = M,
00 Do n ,711=4 m=4
the Bayes solution of the statistical decision problem
becomes asymptotically independent of the a priori probability ot.
Roughly speaking, the condition (6) together with (3)
express the natural requirement that the approximation of M
by the successive intersections of the sets Q ('Wt,) can be
arbitrarily close and the-condition (7) means that the sequence
Z+) -r , 23) ... , exhausts with probability one whole set of po-
sitive integers.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
15
For instance, the condition (7) is satisfied whenever
the integral-valued random variables t,,) CQ)T3)?? are mutu-
ally independent, identically distributed, and such that
6,4, {w : z (w) = mv. I > 0
for NYt.= 3j...
Clearly, under the last condition,
n ioj: -C (CO) *Mill =(Ctlilf -C, (CO)
n=T n
z,(w) M}'< 1,
for 2, 3, ... tyVt = T a. 3, hence,
for, cwt, _4,Z,3,,., i.e.
D
or, equivalently, (7) holds.
Our Theorem 4 can be completed as follows:
THEOREM 5. If o(, > 0 and the conditions (3), (6), (7.)
are satisfied then there exists a positive . integer A such
that (3,n, is the Bayes solution of size /VL, of the statistical
decision problem (c', ) whenever 'n > .
Since by Theorem 3 ~n_c (? )~ ""~ >~- (? ~~ with probabili-
ty one as in --i oo , hence, by Theorem 4 and by the assumption
0 of Theorem 5 it suffices to show that 10,1
i.e. that
(8) ~t,c.w) E A-MIn w: (w)E r1 a(zn(w))) = o?
In order
to simplify the notation we shall write
f1 . L) 4i : -, (w) = ern. I = G.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
FR ~J,~{ I
ILJJun3L'~!J
It follows from (6) that
00
16
Gn {W: ~(w)~ A-Min{w: r-1(w)EnQ(z,~Cw))
hence,
Nc. (G n {w : i (co) E A - MI n {co: nl (w) E nQ(t,~Cw))}) O
and since by (7) 641 (G) we obtain (8), q.e.d.
Let us denote by the decision function which asso-
ciates with every sample point )( from X the decision I
or 0 , according as (1,1) 1~ ....) or X (1)1)1) ....) . By
Theorem 2, is the Bayes solution of the statistical decision
problem (~~ ) with respect to the whole sigma-algebra 3 ,
hence, it is of infinite size. Since the probability of the ran-
dom event of incorrect decisions is equal to zero, the decision
function 3 , in fact, becomes a proof that the value of nZ
belongs to M or to its complement A-- M
Now we shall introduce a function on the values of
which are positive integers or 00 as follows:
{w:#(co)=1}= f ): ~4 )=0},
{w: e(w)- m, {ws ~Gm,Cw) =0, n fl f
3=~
for M. and
C'O
We see at once that # is an ordinary random variable, provid-
ed that the definition is modified in such a way that the possi-
bility
~w{w' QCw) = oo} 7 0
is not excluded.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
17
The random variable .e is said to be the length of the
decision function (
THEOREM 6. If the conditions (3), (6) and (7) are satie-
fled then the length of the decision function (3 is infinite
with conditional probability one under the condition that the
value of 42 belongs to M and it is finite with conditional
probability one under the condition that the value of be-
longs to A- M.
By the definition of
ej 1(&J) Doll
6V C?W: I (W) C M1 fl {w 4,(W f , (z (cv))}).
Using the conditions (3), (6),and (7), we see that .(8) holds,
hence, the first assertion of Theorem 6 is an immediate con-
sequence of (8). Since, in addition,
~w(w: e(w>~.i~~:(w) E A-M})
00
M r) jw: ~Cw)E n Q(2,~(w))J,
the second assertion of Theorem 6 follows at once from (8).
Let us note that under the assumptions of the theorem just
proved it is not true that the conditional moments of # under
the condition that the value of nZ belongs to A- M are finite
i.e. the analogue of Theorem 2 in [72 does not hold. This dis-
advantage, however, can be removed by adding further restrictive
conditions.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
?3. Semantic concepts.
18
The statistical decision problem of ?2 is based on ob-
servations on the sets Q(1)) Q (.Z)~ C~?(3),... which replace
the direct observation on M . The most natural way of how
to get the sets Q(1),) Q Q(3),j is the effect of re-
duction of resolving power in A which can be formally des-
cribed by an appropriate application of equivalence relations.
A binary relation R in the set A is said to be an equi-
valence relation if it is reflexive, symmetric and transitive.
Every equivalence relation in A induces a partition of A into
equivalence sets and vice versa. Two elements jv,(~j of A belong
to the same equivalence set if and only if /It -R cl/ . The equiva-
lence relation s in A is said to be finer than 1 and we
shall write S R if /It S V implies //i. 1 ~/ or, in other words,
if every equivalence set induced by 5 is included in at least
one, hence, in exactly one equivalence set induced by R . Clear-
ly, the set of all equivalence relations in A is partially or-
dered by < and the identity I is the finest equivalence re-
lation.
The formal description of reduction of resolving power by
equivalence relations is intuitively justified by the convention
that two elements of A which belong to the same equivalence set
cannot be distinguished. Under this convention it is reasonable
to introduce the concept of closure MR of M induced by the
equivalence relation P, , requiring that
MR w M{~t,:
Clearly, MI = /vI i.e. the application of the identity I on
M has no effect, and MR C MS , whenever R - S .
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
r P,
.'19
Let RR2, R3, ??? be a sequence of equivalence rela-
tions in A . Putting
MR" = QCn)
for *n= we see that, in fact, the decision problem
of ?2 is based on observations at a reduced resolving power.
This artificial reduction of resolving power is justified by
the fact that, in general, M R has a simpler structure than
M S , whenever R % S .
The application of the elementary facts summarized in ?2
to our main question of statistical estimation of provability
by interpretations in models requires a number of restrictions
which must be imposed on the sets A and M and on the equi-
valence relations R1, R2, R.32
First of all we shall suppose that A is a Boolean al-
gebra. As usual, we shall denote by 0 and 9 the zero and unity
of A , by }2) the complement of the element 1% of A , by A
the operation of forming the greatest lower bound, and by V
that of forming the least upper bound in, A
In order to eliminate misunderstandings we recall that
the subset M of A is said to be a Boolean ideal in A if it
contains the greatest lower bound of any two of its elements
as well as the least. upper bound of any two elements of A one
at least of which belongs to M . The algebraic structure just
defined is usually called dual Boolean ideal. We shall, however,
omit the suffix dual because no other ideals will occur in this
paper.
The relation R defined in k is said to be a Boolean
congruence relation if it is an equivalence relation which, in
410, L,B l '
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
( 1) (';',x., -7
md1
~ttia dDn v... .s
20
addition, satisfies the condition
(9) p. R implies }I' V h ,9 C V,t.
The simplest algebraic structure, which enables the treat-
ment of propositional functions of mathematical logic and for
which the concept of interpretation is. natural, is that of mo-
nadic algebra introduced by P.R.Halmos [2]-
A monadic algebra is a Boolean algebra A together with
an operator V which assigns to every element ft, of A an ele-
went Vp1 of A in such a way that
VII = 1,
V < I,
for every element ft of A , and
d
it. V dcp ) = V It V dcp
for every it and Gy in A . The operator V is said to be
the universal quantifier in A
A subset M of a monadic algebra A is said to by a mo-
nadic ideal in A whenever M is a Boolean ideal in A and
~'L E M implies b'rt E M .
A monadic ideal N in the monadic algebra A is called
maximal if it is proper i.e. M 96 A and it is not a proper
subset of any other proper ideal in A
The relation R defined in the monadic algebra A is
said to be a monadic congruence relation if it is a Boolean
congruence relation and, in addition,
(10) R C t implies Vp. R V - .
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
S ONLY
21
A monadic congruence relation R in the monadic algebra
A is called simple whenever the monadic residual class algebra
A (R)of A modulo R is simple i.e. there is no proper mo-
nadic ideal in A other than that containing the sole element 1.
The relevant properties of closures of monadic ideals will
be expressed by the following lemma:
If N is a monadic ideal and R a monadic congruence
relation in the monadic algebra A then the closure M R of M
induced by R is a monadic ideal in A If, in addition, R
is simple then either M R = A or R
M is maximal.
R
We shall first show that M is a Boolean ideal in A
Let t - E M R and Q E A. By the definition of the closure
M R of- M , there exists an element of M such that
fir, R ! " . By (9) 1r~ V R V 0 i . e. 4Rp.'., hence,
/t, V Cy R 1Z V ?. Since M is a Boolean ideal.in A
,
we have At. V Cf E M , hence 7L V C' E M R . Now let us
suppose that, in addition, Cy E M . Then there exists an
element /t g of M such that itg R C~ . By (9) we have
i t V Aq R v .z ft' v ,t R fL V , hence,
UtLV.1 )'v0R(-kvAq)'v0
vhg)'v0 R(' vq)'v0
or, equivalently,
A/1 A At?
~ AA Rtvn~
and, using the transitivity of R , we obtain
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
AA A t,. R - ,i n
Since M. is a Boolean ideal in A we have AP A xg E M
R.
hence, ,v A E M . We see that M is a Boolean ideal i.n A .
Now it remains to show that M is a monadic ideal in A i.e.
that dr1. E M R whenever ri E MR . Since .t it
follows from (10) that V,&. R d''1, hence, using the assumption
that M is a monadic ideal in A , we have V A ft E M and, con-
sequently, E M R . This completes the proof of the first
part . of our lemma. if 'R is simple then, by the definition
of simplicity, the class of all congruence. sets which have
a non-empty intersection with M is either equal to A (R )
or to the monadic ideal f l in A (P), hence, either MR= A
or MR.= (tl:p q.e.d.
A monadic logic is a pair (A, M) , where A is 'a mona-
dic algebra and M is a monadic ideal in A . The monadic
logic' (A , M) represents a deductive theory in A . The elements
of A:. which belong to M are called provable. If R is a simple
congruence relation in A then the closure M R of M induced
by R is said to.be an interpretation of M in the model A (R).
if ' an 'element' p of A belongs to the interpretation MR of M
we shall say that fl is true in that interpretation and other-
wise false.
The monadic logic ( A, M) is said to be semantically con-
sistent if there exists at least 'one interpretation of M in
a model.
R
Since MC M , we can state that a provable element of A
is true in every interpretation. Whenever the opposite conclu-
sion is possible then the the monadic logic (A, M ), is called
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
u J 11! ~1u 1~~
23
semantically complete. More precisely, the monadic logic (A, M)
is said to be semantically complete if M is equal to the
intersection of all its interpretations.
For our purposes, however, a restricted version of seman-
tic completeness is more appropriate. Let I be a class of
interpretations of M . The monadic logic (A , M) is said
to be semantically -complete, whenever
M-n Q.
QE
In order to eliminate degenerate cases it is natural to
assume that the monadic logic (A, M) is semantically consistent
and semantically 17/-complete. Clearly, the assumption of
semantic consistency can be replaced by M74 A and, by our
lemma, there is no restriction of generality if we assume that
the interpretations from are maximal monadic ideals.
The estimation of provability or non-provability of ele-
ments of amonadic logic is based upon the inspection of its
truth or. falsehood in interpretations in models. Since to each
interpretation Q from there corresponds a simple monadic
congruence relation RQ such that Q = M RQ , the idea of
artificial reduction of resolving power by simple monadic con-
gruence relations is justified by the fact that, by the lemma,
the induced closures are maximal monadic ideals which evidently
have an extremely simple algebraic structure.
The application of the results established in ?2 to the
question of statistical estimation of provability in.monadic
logic requires a further restriction, namely, that is de-
murable. In this case we can write
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
24
f Q (1) Q (2), Q (3), ... } .
The random variable chooses an element.of the monadic
algebra A I the provability or non-provability of which is
to be estimated on the basis of interpretations of M chosen
from 7 by the random variables T1 ~ T2 , ... ' Z'
One may intuitively expect that the following decision
procedure is the most favorable one. At the first step we choose
the interpretation Q ('n.) determined by the value of 2"q . If
the value of 11 is false in this interpretation, the procedure
is stopped and the value of 17 is estimated to be non-provable.
In the opposite case we continue the inspection choosing the
interpretation Q (/M) determined by the value of T2 . If the
value of is false in this interpretation, the procedure is
stopped and the value of /71 is estimated to be non-provable.
In the opposite case we continue the inspection choosing the
interpretation Q (',n) determined by T,3 and so on. Exhausting
all the interpretations Q (,?n) determined successively by the
values of 'r', , T2 , . ? ? , Tn without reaching the decision that the
value of is non-provable we accept the decision that the
value of is provable.
In fact, the decision procedure just described minimizes
the probability of making an incorrect decision only if the
a priori probability ac that 11 chooses a provable element of
A is sufficiently large. Whenever OC is small then the degene-
rate decision procedure which always estimates the value of
to be non-provable is better. The exact discrimination between
these two decisions procedured is contained in Theorem 4.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
25
If the a priori probability oC is positive, if the values
of the random variables T", , T2 , Z3, ..? exhaust with probability
one whole. set of positive integers, and if the monadic logic
(A , M) is 17 -complete then, by Theorem 5, for a sufficient-
ly large number of interpretations to be inspected, the non-
degenerate estimation procedure is the most favorable one in
the sense that the probability of making an incorrect estimate
becomes a minimum. Let us note that the condition of semantic
consistency is in?this case fulfilled automatically as always
whenever CC > 0 and (A, M) is semantically -complete.
In the language of monadic logic the decision functionj8
of infinite size occurring in Theorem 6 is said to be the heu-
ristic reasoning about.the element of A chosen by '!Z and the
random variable ,I is called the length of the heuristic
reasoning 113.
The content of Theorem 6 can be expressed as follows:
If a > 0 , if the values of the random variables 'rt , Z2 , Z3 exhaust with probability one whole set of positive integers
and if the monadic logic (A ,14) is semantically Vj/-complete,
then the length of the heuristic reasoning about the value of
is infinite with conditional probability one under the
condition that a provable element of A has been chosen by
and it is finite with conditional probability one under the
condition that the element of A chosen by was non-pro-
vable.
Clearly, only the last assertion is practically effective
because only non-provability can be discovered after a finite
number of steps. On the other hand, this pessimistic opinion
tl n
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
N
26
concerning heuristic reasoning is. weakened by the fact that if
provability is estimated then this result is asymptotically good.
11 ~r7
;I y
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
ONLY
References.
[31
J.L.Doob: Stochastic processes. New York 1953.
P.R.Halmos: Algebraic logic I, Monadic algebras.
Comp.Math.12,(1955),pp.217-249.
P.R.Halmos: Algebraic logic II, Homogeneous locally
finite polyadic Boolean algebras of infinite degree.
Fundam.Math.XLIII,(1950),pp.255-325?
P.R.Halmos: Measure theory. New York 19
J.Neyman and E.S.Pearson: Contributions to the theory
27
of testing statistical hypotheses. Stat. Research Mem.,
parts I and II (1936 and 1938).
[6] A.Perez: Transformation ou sigma-algebre suffisante
[7I
et minimum de la probability d'erreur. Czechoslovak
Math.J.7,(1957),pp.115-123.
A.Spa6ek: Statistical estimation of provability in
Boolean logic. Transactions of the Second Prague Con-
ference-on Information Theory, Statistical Decision
Functions and Random Processes, 1960,pp.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
O N RAN D O M. OPERATOR EQUATION S
O t t o Han 5
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Introduction.
In the present paper we deal with random operator equations,
in partj.cular?with random integral equations of Fredholm and Vol-
terra type, where only the kernel is random. It can be shown that
such a model is general enough to cover other cases, for example
random limits of integration.
However, the author was not able f.:ive a more detailed dis-
cussion of the subject, mainly because of the lack of-time. Never-
theless, he prefers this somewhere even blurring preliminary version
to a short summary.
Random operator equations.
First of all we shall introduce some basic conceptions indis-
pensable for our further considerations.
Throughout the whole paper we shall use a probability space
(SZ is / ?) with a complete probability measure etc. , i.e. .~ is a
non-empty set, a 6-algebra of subsets of the set.. , and (U.
is a probability measure such that 1t.((Ee) - 0 implies .E E ti
for every E . E o . Further, unless there will be a statement to
the contrary, X denotes an arbitrary separable Banach space and
36 the 6-algebra of all Borel subsets of the space X ..
D e f i n i t i o n 1. A mapping V of the space n into
the space X is called a generalized random variable if
fIw:V(&j)e13 3:BG1}cam.
D o i n i t i o n. 2. A mapping 7- of the Cartesian pro-
dno .Q r Z into the space X is called a random transformation
iZis for every x S Z a generalized random variable,
is
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Two generalized random variables V and are :assumed to
be equivalent if
,atw:V((O)=W((J)
All the classical notions of the functional analysis, like the
inverse operator, the resolvent operator, the adjoint operator, etc.,
are carried over in an "almost sure" way, that is, e.g., the mapping
5 is said to be the inverse of the mapping T if
5 ((j IT (w I Z)) = Z for every z E Z 3 _ 4.
Finally, we recall tL following three theorems on generalized
random variable.e
T h e o r e m 1. If V4, V1 , ... is a sequence of general-
ized random variables converging almost surely to the mapping V ,
then V is a generalized random variable. nn
T h e o r am 2. A mapping V of the space .1[ into. the
space X is a generalised random variable if, and only if, for every
bounded linear functional i e Q , where A is a subset of the first
adjoint space X" which is total on the whole space x , the com-
pound mapping J(V) is a real-valued random variable.
T h e o r e m 3. Let V be a generalized:.random variable
with values in the space X and T a random transformation of the
Cartesian product .l x X into the space Z that is almost surely con-
tinuous. Then-the mapping W o(' the space 1 _ into the space Z
defined for every W e 11 by the formula
W (w) = T(&, V(w,)
is a generalized random variable with values in he :,p^ce
z
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
4 -
Let X and Z be two separable Banach spaces, T a mapping
of the space X into the space Z
z
, and Z a
. If 7- denotes the set of those elements
the equality T('c) Z holds, i.e. if
then any x E,L is used to be called a solution
tion
fixed element from
x E X for which
of the operator equa-
T(') - z /1/
If the set is empty, we say that the operator equation /1/
does not.possess a solution; if it is non-empty, we oay that /1/ is
solvable. In the case E consists of exactly one point we say that
/1/ has aunique solution.
Now, let in addition be a probability space with a
complete probability measure ? and X and 7 be the 6-algebras
of all Borel subsets of the space X and Z respectively. if 7
is a random transformation of the Cartesian product x X into the
space Z then
T(.,; ) = z
/2/
is said to be random operator equation.
Hove*'ver,-the relation /2/ does not express the most general
form of 'a random operator equation, namely the right hand side of
/2/ need not be a fixed element from the space Z , but can be.re-
plaoed by a generalized random variable. with values in the space
ore?v'er, it should be remarked that the solution of a random ope-
retor equation does in general depend on the choice of WE-f2 ; , con
oGgmontlyg the most general form of the random operator equation
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
may be more precisely written as
T(., z(.~ /3/
where, as mentioned above, T is a random transformation of the
Cartesian product fl x X into the space Z and Z is a general-
ized random variable with values in the space Z
Similarly as in the deterministic case, with the only except-
ion of neglecting a set of probability measure zero, the wide sense
solutions of the equation /3/ are defined, i.e. every mapping W'
of the space #1 into the space X satisfying the equality
T(w,ir(w)) z N)
for every .6) from a set Ile of probability measure one is said to
be a wide sense solution o? the equation /3/. However, following
th't spitit of our previous papers, it i? quite natural to require
the condition of measurability to be fulfilled in order that we can
speak about random solutions. Thus, ifthe ride sense solution is
moreover measurable it will be called the random solution, a we
,can state
D e f i n i t i o n .3. Every generalized random variable .Z
with values in the apace satisfying the condition
Taw X (")) (w)
will be called, the random solution of the random operator equation
/3/:.
Evidently, there may exist wide sense. solutions that-are not
random solutions. moreover, if the random operator equation has
more than one solution for every W from a set of positive pro. a-
bility:'"measure then they may be, in dependance on the - alFebra
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
of course, many wide sense solutions that are not measurable. As a
simple example of this fact let us mention
8 z a m p 1 e 1. Let X be the apace of all real numbers,
E anon-measurable subset of the space a and T a random trans-
formation of the Cartesian product n.x X into the space X , de-
fined for every W E and x e X by the formula
T(t,,x) : A: - 4 .
Then the mapping 'r of the space n into the space X de-
fined by the formulae
ir(u) = for k) C -E W- (i) a -1 otherwise ;
An a wide sense solution, but is not a random solution, of the ran-
dom operator equation
T(?, f(.)) _ 0 /4/
Roughly speaking, we are therefore interested mainly in the
case when for every W CE , nnL there exists a unique solution of the
deterministic operator equation
TCw, ) = z Cap)
More precisely, we.shall investigate-most frequently the case when
'here exists a unique wide sense solution, provided we identify +-'
solutions differring on a set of-probability measure
Nevertheless, even under this restriction, the uni
Be solution need not be measurable as shown by the following
E x a m p 1 e 2. Let f. be the space of all real numbers,
the 6--algebra of all at most denumerable sets of real numbers
and their complements, and & a complete probability measure defi-
ned by the formulae
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
re)r~
aka
al_
_. t
J ~A w,a EJ1'A~1~j
(E) = 0 if E is at most denumerable;
(E' 4 if the complement of E is at most
denumerable:
Further, let X be the space of all real numbers with the 6-Algebra
of all Borel subsets, and T a random transformation of the Car-
tesian product 129 X into the space X defined by the formulae
7- (1W, A) = 0 for w - x ;
T(vr, x) other'Nise.
Than the unique wide sense solution of the random operator e-
quation /4/, given for every 4j a fl by the formula
Y(w) ? 4)
is not a random solution, because of the fact that, e.g.,
ftj:W(4J)1. 0 3 ~ .
`any other questions concerning the relationship between wide
sense solutions and random solutions of the same random operator
equations arise, the greater part of them having been as yet unsol-
ved. Unfortunately, we too are not able to present some useful theo-
ry of random operator equations unless some further assumptions are
imposed.
In the present paper we shall discuss mainly three particular
cases, namely if yy
a/ the separable Banach space equals to the separable Banach
space Z ; and/or if
b/ the random transformation T is almost surely linear and
bounded.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
It is not surprising that in the theory of random operator
equations the main role is played by probabilistic v-,rsior.~i of the
well known Princil i?, of Contraction l appinj :. at ' its many modifica-
tions and generalizations, which under appropriate assumptions `'ur-
nish the existence, uniquene.a, and measurability of the random solu-
tion of a random operator e-.uation.
The foilowini, theorem is a useful st ;rting point for other theo-
rems of this kind.
e o r e m 4. Let T be an almost surely continuous
transformation of the Cartesian product.Qx X into the spa-
c. X so that
~{~ IIT"`(~,x)-T"`(u y)N?(1-,> ).Vx-. H1~='f ; ~/
C/Cu .t( nit
A%04 %041 "X
where. for every (Jell , x e -na n 4 1,
T.'q(WJ* K) - T'(w x)
and
T`4~w~x) T(w~ T"(y x
Then tt:ere exists a generalized random variable with values
ce X satisfying the relation
(" {w: T(``i~ ~cw)) a ~(~~ J - 1
L:ore,jver, if there exists another generalised random variable
with the property
,U [w : T (4),T (Q) = IP(()) = ~1
P r o o f. Let us denote by E the set of those w from the
set occurring in condition /5/ for which the mapping T((J,-) is con-
tinuous}. Svidentiy, according to the assumptions, Now,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- o -
let us define the mapping 0 of the space fl into the space X so
that for every WE E the point +(w) equals to the unique fixed point
of th i mapping TO), ') and for every w 6 n - E we set 4(4j) = 9, where
8 s the null element of the Banach space X Thus, relation
/6/ holds.
In order to prove the measurability of the mapping 4) we make
use of theorem 3. and theorem 1. The remaining statement follows im-
mediately from the uniqueness of the fixed point of the mapping Trk;')
E
for every 4)6
The just proved theorem forms a generalisation of author's pre-
vious result, the stronger assumptions of which enable one to for-
mulate condition /5/ in a amore tranenRrent way.
T h ? o r ? m 5. Let T be an almost surely continuous
random transformation of the Cartesian product A X X into the
space X and G a real-vzlupd random variable so that the follow-
ing relations hold :
[1j: C CO) ~ 1 J - I j /7/
7--(&),x)-T~wy)II C (w) IIz-III _'1 /8/
for every two ele:-,,cnte x and ' from X
Then there exists a generalized random variable 96 with va-
lues in the space X for which 'relation /b/ holds.
P r o o f. Since the Banach space X is separable, we can
replace the intersections in the expression
n l l t :IT(w,x)-T(y,y)~~c(y~ Ax-y q~ {w:c(b)c~l~n~w'T(n,?) is continuous
xEX 36%
by intersections over a countable dense subset of the space X and
what proves our theo-
thus condition /5/ is fulfilled with 11='I
rem.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
R e m a r k 1.. It should be remarked that both theorem 4
and theorem 5 can be reformulated and proved also in the case the
space X under consideration is a metric-space. However, in the
present paper we formulate all the theorems for Banach spaces only,
though the "metric" versions of theorems 4 and 5 are used in proofs
of theorems 11 and 12 in order to avoid more complicated consider-
ations.
As an immediate consequence of theorem 5 we get
T h e o r e m 6. Let C be a real-valued random variable
and T an almost surely random transformation of the Cartesian
Product 11 x X into, the space X satisfying the condition /8/.
Then for every real number -X $ 0 such that
,u.{w.C(w) < I.Al 3 -I
there exists a random transformation 5 that is the inverse.of
the random transformation ( T
P r o o f. Evidently, as .. $ 0 , the random transformation
is invertible whenever the random transformation (
is invertible. '?owever, for every z E X the random transformation
T defined for every W E and X E by the formula
4
T (4j, x) - z
is almost surely reducing and therefore by theorem 5 there exists
a unique random fixed point it, satisfying the relation
(U. L lJ ' Y~ (6)~ _ .~, T(W, x; (w)) - Z 3 ='1 .
Since the last statement is equivalent to the invertibility of the
random transformation theorem 6 is proved.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
am 11 a.,at:-,r
It is a'well known fact that any linear bounded operator A
s%tisfies the Lipschitz condition with the conata1t IIAI~ , what is
at,the same time the smallest constant with such a property. There-
fore making use of this fact and some classical results about li-
near bounded operators we can state
T h e o r e m 7. Let T be an almost surely linear bounded
random transformation of the Cartesian product fl x into the spa-
ce X ?
Then for every real number2 40 such that
U 11 T"
there exists a linear bounded random transformation 5 that is the
inverse of the random transformation (T and we have
~u (~1 {(j: s cw,x - 71 1 T (w, X) })
:eX 4t. p
where the sum is meant uniformly.
A number of interesting theorems on random integral equations
has been derived by A.T. Bharucha-Reid. Here we mention theorems
2.1 through 2.3 only, which can be a little strenghtened using theo-
rem 7, namely we can write
T h e o r e m B. Let T be a random transformation of tka?:'
Cartesian product -~ L x X into the space X which is for every.
E [ linear-and bounded.
Then for every real number J.. # 0. the set
.~2 (..) w: ll T(~, ?) I1< !al }
belongs to the 6-algebra , the random transformation
is invertible for every W the resolvent operator R:x r exists
for every W,6 and for these W's we have
Ra/T As./
Finally, for evay W E.~(a) the solution S(4)) of the operator equa-
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
.)f =z
is for every z e X given by the formula
S (u) = R, T N' z) 1
where the resolvent operator Ra,T and consequently the solution
,S as well)is measurable with respect to the 6-algebra 0(..)n
P r o o f. Theorem 8 follows immediately from our preceding
theorems and from well known classical results, because of the fact
that for every U 6 ri
where X ~ .X~, J ? is a countable dense subset of the sphere fs- ixij 1 j.
Heretofore we have always used the above formulated assumption
a/. sometimes together with the assumption b/. Now, we shall state
two theorems in which the spaces X and Z may be different separa-
ble Banach spaces, provided the random transformation T is almost
surely linear and bounded. The proofs are omitted, because both the
theorems are only slightly modificated previous author's results.
T h e o r e m 9. The inverse of an almost surely linear
bounded invertible random transformation T of the Cartesian pro-
duct .I x X into the space Z is a random transformation of the
Cartesian product flu Z into the space X ..
T h e o r e m 10. Let X and Z be two Ban$ch spaces
;',hose first ad3oint spaces X4 and Z are separable; let X- , 3
and 7* be the 6-algebras of all Borel subsets of the space
x and z respectively. If T is at almost surely li-
Z
near bounded mapping of the Cartesian productn 1 2% X dnto the spa-
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
13
ee Z then the following two conditions are equivalent :
for almost all elements W E.Q the mapping 7_(4J)*) of the space,
X onto the space Z is invertible ;
for almost all elements. 4)E-.Q the range of the adjoint mapping
T7w,?) is the wkole space
Further, if these conditions are satisfied then T is inver-
tible and the inverse mapping (T*)to the adjoint mapping T* is
almost surely equal to the adloint mapping S of the inverse to
the mapping T* S
Moreover, if one of the mappings T , 5 , , is a ran-
dom transformation then all four mappings are random transformations.
One of the important problems in the theory of random operator
equations is the question of the measurability of the solution, which
has been dealt with in the preceding theorems. Now, we shall be con-
cerned with the question of the relationship between the random so-
lution of the random operator equation and the solution of the cor-
responding deterministic operator equation. More precisely, we shall
discuss the case when the random operator equation /3/ is such that.
the Bochner integrals
and
2M d? (w) - z
S (4)
exist for every J( 6 X . Let us assume that the solution of the de-
termini?tic operator equation
5( ) Z.
equals to The question arises whether the expected value of
the random solution of the random operator equation /3/ exists, and
if so, whether it is equal to the deterministic solution
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 14 -
It is not difficult to construct an example showing that there
are cases in which the answer is affirmative. A most trivial one is
that when the probability measure (. is a Dirac measure, i.e. when
there exists an element W0 E n such that 4 (4)0) ? 4 . Another
still trivial example is the following
E x a m p 1 e .3. Let T be a random transformation of the
Cartesian product -a x X into the space X defined for every 4! En
and .Z. E X by the formula
T(",.) - c.* + You)
where C # 0 is a real number and V a (:eneralited random vari-
able with values in the space X so that the Dochner integral
SnV(W) dre(w)
equals to the null element of the space X
Then the expected value of the unique random solution of the
random operator equation
T(, ()) e
equals to the solution of the operator equation
5(1) = 0
where, of course,
S ( ac ) = c.X
for every x 6 X
It is also not difficult to give examples when the answer to
the above stated question is negative.
However, there are many cases in which we are interested in the
solution of the deterministic operator equation dorrespondinzg to a
given random operator equation rather than in the expected values
or even in the probability distribution9 of the random solution ok
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP8OTOO246AO11700340001-4
the random operator equation under consideration. Really, the case
of determining LD 50 is one of them.
Let us call, for the sake of brevity, the deterministic opera
for equation associated with the random operator equation by means
of expected-value correspondence simply the regression operator e-
quation.
Thus, an important problem of the theory of random operator
equations is that of reaching the solution of the regression ope-
rator equation, and it is this problem the remainder of this section
is devoted to.
Since a detailed case history of this and similar problems is
given in a coon paper written by U. Driml and the author which
will appear it. the Transactions of the Second Prague Conference on
3L,f4rmation Theory, Statistical Decision !unction, and Random Pro-
ceases, we shall not go into details here, bust shall state three
useful theorems orly. Nevertheless, it should be noted that this
'branch of j: robability theory, often called :ssporience theory, is
very tightly connected with stochastic approximation methods ad de-,
veloped by H. Robbins and S. ;.;onro and other. authors.
The followinr theorem, as yet unpublished, is due to M. Driml
and the
author and can be
~orienoe theory.
T h e o r e
cede mapping; t1e
X
and r,
counted to the basic theorems of-tre ex=
Let r
be a generalized stochastic pro-
Cartesian product -a xEO,?O) x X
into the space
surely continuous with respect to both the argu
netts 0 and xE
~iinultaneously. Let there exist an element.
..x X , a real-valued random variable 0 , and let the following
conditions, together with condition /7/, hold :
Approved For Release 2009/07/09: CIA-RDP8OTOO246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 16
t
{w: II t f -rcws-X I1 = 0 3 _ ~I) /9/
ZW' IIT(w,f,z) - T(w,f, )11 `- C(to).11-A - 113
for every f > 0 and every two elements J6,1 G X .
Further, let J () be the solution of the random operator
equation
toy) = T(-)Djlo(-))
t /10/
JtC?) ? t f TC stt )) ds for t > 0 ,
Then .xt is for every f ? 0 a random solution of the ran-
dom operator equation /10/ and we have
(W: x (Q) is continuous in +~
and
?{,w:ti-m, Ilx.tcw) - 11I=0} =
P r o o f. Let us denote by C the space of all continuous
mappings f' of the space (0, oO) into the space X such that
the relation
11Cf)-tll 0
T ?p
holds. Introducing the 3H.stance function for every couple of
elements 5, cj E li by the formula
to p 504
the space C becomes a separable metric space whose 6-algebra of
all Sorel subsets is the 6"-algebra generated by the class
further, let us denote by 5 the operator on the Cartesian product
n x C defined by the formulae '
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Cs(w,f)] (o) = T(w,o, f(o))
and
[ S (w, ~~ ] (f) = { JT'(w, s, 6cs~) ds
for every E C , t > 0 , and every w a E , where E equals
to: T'(w,t,z) is continuous in t and .x simultaneously] n
n Ica: CM0 let us put
[S(4),f0(4) - .x .
First of all let us prove that the mapping S maps .j x C into
C . Choose arbitrarily we E , f E C , and 4 . Then tho
re exists a real number t, such that for every f ~.
I f(4) - 211 I7 %
and simultaneousl.
Then for every
can .rite
t
ST(w,s,fhs))ds -x a
T(w,sf(s))-T(w,s,x)I1cis +I! JT((,3,s,x) iIJ
0
t
0
tfi C(w) () + C (W 1 {"t ? S~,(,p Il f(s) -.X u + %
t.ss?t
JT(cj, s, x) 4 - x
?
0 it
/13,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Since the (.-E) part is trivial we have
C proved that S maps the
Cartesian product -a n C into the apace C
Now, let us prove that S is a reducing transformation. Thus,
let us have ~, g E C and WEE / for W eD-E we
get a singular
case/. Then
r (S Ca, f) t S (w, 9 )) -
sop ? 9 t f T(Q, S,f(s)) d S - /T(4),S,(S))dAs
t
IR $sI II T"(4) s~ ~c 5~) - T(~ s, 9 try) ds
C('))? (~,g)
The mapping S being a reducing random transformation of the Car-
tesian product n x C into the space C we can, making use of
Remark 1, ai:;:ly theorem 5 which asserts that there exists a general-
ized random variable with values in the space C so that
/4. {4: S(w cp(w)) _ (p[w)~ = 7
However, we have E 6 C , (E) hence setting for every 0El2
and every f > O
x{cw~ = t~(w~J~~~
we get immediately all.the assertions of theorem 11.
A generalization of the preceding; theorem for almost surely li-
near bounded random transformations is the following
T h e o r e m 12. Let T be a generalized stochastic pro-
cess mapping the Cartesian produet.-n C X C0~?p~ x X - into the spa-
ce X and almost surely continuous pith respect to both the argu-
ments 0 and it E X simultaneously. Denote by the mapping of the Cartesian product x X into the space X de-
fined for every C0E..Q , .4 e X , and / - 1 2 . by the formulae
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
~fr.':'r 19
.and
T: (4) ) (w/
x).
T. Cwx) - - T(w 1,, T Cw.)
Let there. exist an element E X so that condition /9/ and the
following oonditioni are satisfied':
fit. { w . T(4; i ) a4.x + ( j ) = oc T(w,i, x) t (S.T(w, t,~) 3 _ ~1
for every f > 0 , *,) e X , and any real numbers of and ( ;
oldJ(J: Sup I7(4 tj-)q,CQo}
~ " *nn jo . 11 T
kLq u n
f ^-4 A&I S*X i.1?-; 70 0 i' ) '
Further, let X{ () be the solution of the random operator equation
/10/.
Then ?X,6 is for every f ) O the unique random solution of the
random operator equation /10/ and the relations /11/ and /12/ hold.
P r o o f. The proof follows that one of theorem 11. :e mlAt
only replace the inequality /13/ by the ingquality
3. { P If T(w, t, .) JI . k
and prove the contraction property in the following way :
Denote by 5 *"the mapping of the Cartesian product -a x {.r+
the space C defined for every 4 J , 6 Se C , and, "t = 4, 2, .. .
by the formulae
S~(w/S) = S (w/ ~)
and
Sfit+4 5 (4j /4))
Then we can for every S c- C , every nt z 4,2/ ?.. , every > 0
and almost all W 's write
C5'~c4~, )] J) [s
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-20-
~1,T, t.
G l
-S 4 ~ L5M-A(WI f) J (S,)1
j ~
J 1 t s,
a f T `Q, si , s~ J T (w, S1 Es(w T) js2))
0
t fS4
0 4 T(w,s4,T(w,s: [s"'~lw
o , C , )](5o)) dsz ds,
ft S,
,! S S,)si (W, Cs$)J (Si.)) d.S~ ds,~
0
and hence by induction
4 S' S-
101 S f 4 0 o e
Thus, for every W from a set of probability measure one there e-
xist positive integers 'n, and n'- such that for any two elements
~,CC we have
,4))
S(3S(40,3)) S,(4j
t S.
1
tso1 t J S, ,~ ? .. S,-, s,...~ (W)S(s,,) - U.)) ct
c 0 ~
and hence applying the "metric" version of theorem 4 we get the de-
sired result.
Theorems 11 and 12 and many other theorems of the same type
claim the fact that under appropriate assumptions the "decision pro-
cess".converges to the searched fixed point of the regression trans-
formation with probability one. However, in practical situations
the statistician scarcely knows whether all the necessary conditions
are. satisfied. A kind of justification of his decision about the
fixed point of the regression transformation is contained in the
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
following
T h e o r e m 13. Let T be a generalized stochastic pro-
cess of the Cartesian product Q it [0,00x into the apace X and
x a fixed element of the space X . Let A,() be the solution
of the random operator equation /10/ and let the following relations
together with relation /12/ hold :
[4): t m. If1((j ,f,x~(0)) - T(wJf,, ) I1=off
-0 00
t
I N : S I T-(w,s, -1,(w)) -r (w, N.d.s < oo s 1
for every f >~ O .
Then the relation /9/ holds.
P r o o f. According to the condition /12/ it-suffices to pro-
ve that t
{w : 1 t I a f T(4),s,2) an ' X (w) II = o 1
t
r(11,S,x~ t
4. f T(w,s,x~c~)) II=o}a ~.
d
However, the last relation follows immediately from the trivial in-
equality t t
u t j T (W , =, x) ~~ - t J T-cw, 5, x (w)) OtS
o p
it J ~I T'(', s,x) _ T-(wis / ,xs (u)) II d.s
O
which ho13a with probability one.
Some other theorems dealing with experience theory problems
can be found in some papers published in the Transactions of both
the First and the Second Prague Conferences.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
22 -
~Oj
Random integral equations,
In the remainder o.: this preliminary v"r-..ion we shall briefly
deal with a special case of random linear operator equation,, name-
ly with random integral equations.
In this section X denotes the separable Banach space of a!)
continuous functions defined on a closed it t erval w:;ere
Je a. , with the norm
#JON rr o..( j.xcuJ 1 .
*I g4.
Further, let us denote by the set
f(u,v) 4s+~L4JA.iY~
First of all we shall rcc:_11 tne well tno'.vn r -::ult from fun-
ctional analysis, namely
T h e o r e m 14. If A (a,,r) is bounded for every (ir,v.~ E a
and, if all aiacont?.r>>ity ;:oints of 4 are situated on a finite num-
ber of curvpr
"4,2,..,lt.
where the functions are continuous, then the formula
4.
defines a.compact linear operator on the ooice X Into itself.
For the sake of simplicity. we shall not work with the most ge-
neral form of the kernel A , but shall assume only kernels all
discontinuity points of which. are situated on the curve 1V' = u.
It woui be still simpler to discuss continuous kernels only,however,
this would exclude the important class of integral equations of Vo,---
terra type.. Therefore we have accepted the above stated coniromiz,
Thus, let us denote by f'( the space.of all kernels it all
discontinuity points of which. are situated on the cure' V- =,u-
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Introducing the norm given by the formula
!l~,l! = sup/4(u7J/)
where the least upper bound. is taken over the net
the space K becomes a separable Banach space.
Now, we can state the following two results concerned with;the
measurability of the map; ing T occurring in the theory of random
integral equations.
T h e o r e m 15. Let the mapping T of the Cartesian pro-
duct , L x X into the apace X be defined for every Well and
by the formula
T(,ij,, x) J (w~ ', V-) -t (t') ', /14/
where IG is a mapping of the space into the apace K such that
for every and every, real number, /t .
:?:en the mapping T is a compact linear random transformation.
P r oo f. That. the mapping /14/ is for every ii Efl linear
and compact follows in .attiately from theorem 14. Let us therefore
prove that it is a random transfor:ution.
Thus, choose an arbitrary X Cr x . According to theorem 21t
3uffice3 to ;rove the measurability for every X e X and. every 1i
near bounded 'functional defined for every 4,t, a C4t,tJ and' e '
by the formula
that is, it suffices to prove the measurability of the mapping.
,6-
~4 ET?w,x)J = f (.'d,v-
]L____.~_ . ..__ Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 24 -
for every fixed I4 E r4,&1 and X E X . However, by the sane theo-
rem 2 we get that the statement
is a real-valued random variable for every (4,v) E Q and X C- X im-
plies the statement
.k(?,4,?).4(.)
is ? )-er.eralized random variable with values in the space X*
! owv,? the mapping associating to every WC-1Z , every kernel' 4' , and
every jc 6X the real number
A (w, 'cc. rr
is a linear bounded functional, say $4, , on the apace X , so that
E T(A)
dOA1 [4 (ej, u,
and thus ac,---ding to theorem 2 T is a random transformation.
T h e o r e,m 16. one compact linear random. transformation
T defined in theorem 15 satisfies the relation
C. B
for every Borel /,vith r^spect to the normed topology/ subset 6
of the space of all ender.)rphisme on X into itself.
P r o o f. Heretofore we have.dPp ht with the random trans-
formation T as with the map;:ing of the-Cartesian product f2 x X
into the space X )nly. ?dow, we shall use another point of view.
Let us denote by Al the space of all endomorphisms on X in-
to itself, i.e. the space of all linear bounded operations A on
X into itself. Introducing the usual norm into this space by the
formula
i/A ll _ 5 ,gyp 1/4 (x)!1
Approved For Release 2009/07/09:'CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
25
the space M becomes a Barth algebra. Denote further by 7A the
6-algebra of all Borel /with respect to the normed topology just
introduced/ subsets of the space M . Thus, we have to prove the
relation
If W:T(ej,?)e8]:BEd?t1cS.
However, this relation is equivalent to the relation
f{J:T(1,,?)e5j :8E 9'3 cC,
where the 6-algebra ' is defined by the formula
[8nF : 8 E 2&J
and F is the space of all possible compact linear operators form-
ed by means of kernels from the space f( , i.e. in symbols
(L : L () J4(?,V) CIPI d r ,~G X 3 E /(J
Because of the inequality
Ill_I wap /L(o)ll =
490%4 #A#- f
m. ax su.An (d - 11 ~t Il . Il.x /l a
a~tfF A~.ri.~ t
and the separability of the space K , the space F is separable
in the normed topology. Since F is at the same time a metric spa-
ce in the topology considered, it is perfectly separable. Therefore
the class of all spheres contains a countable subclass that is a
base of the space F . Hence and from the validity of the relation
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
- 2d -
Ili-L0 I'. # } }
00
T-(6 fL:l/,L-L,l/<
ALS4 .q
for every E > 0 and L. E F , where zl,''r:l .,: is a countable dense
subset of the space X , we get the statement of our theorem.
Similarly as in the deterministic case we associate to every
random integral transformation Y a subset of the Cartesian pro-
duct n Y. [ '?, 010. denoted by (T) and called the resolvent set of
the r::ndom transformation T . T. the measurability of 11 -sections
of the resolvent set c(T) is devoted
T h e o r e m 17. Let all the assumptions.of theorem 15 be
fulfilled.
Then for every real number the set of those !,J's for which
the linear random transformation (T .~ I) is invertible belongs to
the 6'-algebra (, , i.e. for every real number
(T)} E C'3
i' r o o f. According to the well knofa theorem on resolvent
.,sets in Banach algebras the set
g~ a ~gAEM,.~. E;pIA)~
is for every fixed .X open and therefore from the 67-algebra 2
Fence,, theorem 17 fellows from theorem 1G.efid.,the relation
1 4,:(4j,A) e Q(T) } _
f4v:7(47,?)E8a
which holds for every fixed real number ,..
AL oufficL:nt condition for the invertibility of the random
transformation (7--a r) is given in
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
T h e o r e m 18. Let all the assumptions of theorem 15 be
fulfilled. Let in addition the real number -A* 0 satisfy
(GC.t&,:({r-a).Il,~w~?~"~I1" co
where nR = Card {i: pi a K(O) R)) and ~1R is the average of those
variables -9(pi) for which pi is in K(O, R).
Let us introduce two core definitions. Given in a
plane a sequence of circles Kl, K21 ... with radius a/2, the
centers pl, p2,... of which have a given density d, we will
call a mean covering, in short C', a limit
(3.12) C
1im l2 E I Kin K(O, R) I
R >oo TrR i
and a mean double covering, in short C", a limit
(3-13) C11 = lim 1 E E Kin K n K(0 R)~
R->co TrR2 i j
In this sum, the case i = j is not excluded. Of
course, the mean covering C' of the circles K1, K2,... and
density d of their centers pl, p2,... are related by the
equality C' = IK1Id. As a consequence, we may use C- as our
measure of density of centers in comparisons where IK11 is.
kept constant.
Now it is clear that for stochastic processes r(p) with
correlation functions given by (3.5) the search for a sequence
of points with a prescribed density u ich yields the minimum
limiting variance is equivalent with the search for a
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
corresponding sequence of circles yielding the minimum mean
double covering. Moreover, if thane exists such a sequence of
points which would realize the minimum limiting variance for
all positive values of a in the correlation function given
by (3.5), then it would be the best sequence also for a process
with a correlation function given by
00
(3.14) P(u) = f r(u/a) dF(a)
0
where F(a) is a distribution function with F(O) = 0.
It is of considerable interest to find out which
correlation functions admit the representation given by (3.14).
Unfortunately, there do not exist sequences of points which yield
minimum mean double covering simultaneously for all values of a
in (3.5), as will be shown later on in this paper.
ktimal nets of oints
.
ti
Consider a sequence of congruent circles K1, K2... with
centers pl, p2... respectively. Let us denote the indicator
function of Ki by ki(p), that is, let us put
1 for p e K.,
(4.l) ki(p) = 0 otherwise2
Moreover, we put
(4.2) k(p) = E ki(P)
i
In other words, k(p) is equal to the number of circles covering
p. In terms of these functions, the definitions of the cnaar
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
covering Cf and the mean double covering C" given in section 3
take on the forms
CI = lim 1
(4.3)
R-> oo 7TR2
= lim
R--o- co
E I ki(p)dp
i K(O,R)
1 E
7rft2 K(O,R) i
ki(p)dp =
= lim 1 ! k(p)dp;
R-,. co nR2 K(O,R)
(4.4) C" = lim 12 i z f ki(p)kj(p)dp
R es `rR j K(O,R)
f
f
,R)
K(O,R)
E X
ki(p) kj(p)dp
ij
= 11W
R-' 00 irR
2
=lira
R- co
k2(p)dp.
Our problem is to determine such sequences of congruent
circles with a fixed mean covering for which the mean double
covering attains its minimum. We are now going to prove an
inequality from which it follows that a sufficient condition
for a sequence of circles to have this minimum property is that
the set of values of the function k(p) consists of two consec-
utive integers. It is the content of the following.
Lemma ka. For any sequence of circles with mean
covering C', the following inequality
(4.5) C" > (2[C'] + la Cf - (C+) ([C'] +1)
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
i:i
holds, where [C'] is the integral part of C'; equality holds if
k(p) takes only the values (C'] and [C'] + 1.
Proof. Clearly, we have
4.6 l k2(p)dp = 1 l f k(p)dp}2
( ) 2 f WR2
K(O,R) K(O,R)
1 l }2
2 f k(p)dp - (C'] - 2
nR K(O,R)
- dp
+ 12 (k(p) - (C'] 2
.rR K OAR)
As always,
(k(p (4. 7) ) - (C')
we conclude that
2
k(p)dp
f k2(p)dp = r 12 fR)
(4.8) R2 nR R(K(O,R)
2
~- 2 f k(?) dp - (C' +
K(O,R)
For R --?-- 00 , we get
2
(4.9) Cu > Ct2 - { C' - [C'] 1 +
and this is an alternative form of (4.5). Now if (C'] and
[C'] + 1 are the only values of k(p), then (4.7) and consequently
(4.5) become equalities. This proves the lemma.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-
15-Wk-will now describe some sequences of circles minimizing
the mean double covering. We confine ourselves to the case where
the centers of the circles form nets composed of congruent
figures such as triangles, squares, etc. This-will enable us
to compute the mean covering and the mean double covering, from
a single mesh of a net and we shall exploit this possibility.
,The minimum property will follow by our lemma, as the function
k(p) will take only tuo consecutive integers as its valuesAn
our examples. We arrange these examples by increasing values of C'
Case I. If
Ct < = 0.907 ...
2 %,3
Chen the net of equilateral triangles) has the optimal property.
This situation is illustrated in Fig. 4+.1 and Fig. 4.2.
Fig. 4.1. Ct < y -*'d'907...
2;V
1
The pattern may just as well be referred to as a "net of
rhombuses."
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
.~a
-16-
4.2. C'
Fig.
213
...
0.907
For the purpose of illustration, we will present hese the
details of the computations of C, and C".
In Fig. 4.1, we put the radius R of the circles equal
to 1, and the side of the triangle equal to a ; 2. Thus, the
area of the triangle is
A 4 ?
The circles divide this area into four parts
A=T0+3T1=Ao+A1 1
The meaning of To and Ti is shown in Fig. 4+.3.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
FnP,
RC l~J J' _. T. '
U
Fig. 4.3.
Thus,
In T0, k(p) = 0, mile in T1, k(p) = 1. Now
C ' _ 1 { 0 dp + dp) as 1 dp.
JA A A
0 1 1
C1 : 4 (3 b ) = 2r
82 82 ?
For s > 2, C< it/21T3~ 0.907 ... while for s = 2,
v/2 0.907 Moreover, C" = C + as
C" 02 dp + f 12 dp ) = C10
JA A
0 1
We will now compare this value of C" with the corresponding
value of C" for a net of squares having the same value of C?,
and therefore, representing the same point density.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-18- .
We put the side of the square equal to x.
four circles with radius R 1, we get, say
Fig. 4.4.
Drawing the
C 1 odp+f 1.dp+ f 2dp ) a 2.
x
A 0 Al A2
From C1 R/2 - 3 , we get
x2 12; x 2~ = 1.86 2R
If A2 stands for the area of that portion of the square,
where k(p) = 2, we get
A2 = 8 (
360- ,r - , y , 2 12 ) = 8 (3~G 7 - I sin vM
v and y having the meaning indicated in Fig. 4.5.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
J. 1
Fig. 4.5.
Carrying out the computations gives A2 - 0.14. Clearly
Al - v - 2A2 - 2.86. Thus
CI --1-12.86+2 ? 0.14) -0.907...
12
(as it should be), and
coo - 1
_ 12
Case II. If
(2.86 + 22 ? 0.14) - 0.988 ...> 0.907...
0.907... '
2V3
- Cf < - 1.209
313
then the net of equilateral triangles is still Optimal. Sao
Fig. 4.6.
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
-20-
I
Fig. 4.6. 0.907 ... . I < C+ < 31 + 1.209.
However, in this case k(p) has three values: 0, 1, and
2, so that our lemma 4a does not apply. The optimality of the
net in question is a consequence of a known inequality (Toth
fej as.[4J, Chapter III, paragraph B. inequality (3), p. 80)
from which it follows that among all convex hexagons of a
given area and circles of a given area, the maximum possible area
of a common part of a hexagon and circle is reached, if the
hexagon is equilateral and the circle is concentric with it.
The statement of case II follows, if we apply the quoted
inequality to cells which are formed by attaching each point of
a plane to the nearest circle center. The above-mentioned
inequality is of its greatest interest when the mean covering is
in the range indicated in Case'II. Let us note, however, that
it implies also the statement in Case I to the effect that the
circles should be disjoint. /
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 . Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Case III. If
. 27r C' < 2r
1.684...
1.209 ... < 31x3 - 2 +lV 3
then the net of isosceles triangles is optimal.
This is seen as follows. We start with the situation
drawn in Fig. 1.7.
Fig. 1.7. C' _ ? 1.205 ...
3113
We then increase the mean covering without spoiling the property
of k(p) of having only two consecutive integers as values,
letting the base of the triangle diminish and the height of
it csiargo, so that the three circles intersect still in one
point. We can continue this procedure until the length of
the base becomes equal to the radius of our circles, as shown
in Fig. 4.8.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Fig. 4.9. CI
Case IV. If
2Tr
1.684 ...
3
2 +
1.571... 2 C' ~7 = 1.814...
a net of rectangles has the optimal property.
We start with a net of squares and circles intersecting
in the centers of the squares as indicated in Fig. 4.9.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
Fig. 4.9. CI = 2 = 1.571... .
This corresponds to C' = rr/2 1.571 ... . We then
enlarge the mean covering by lengthening two sides of the
square and shortening two others, while the circles still
intersect in the middle, as shown in Fig. 4.10, where the
extreme situation is drawn. The radius of the circles is then
equal to the shorter side of the rectangle. This corresponds
to C' =ir/-J3=1.814...
Fig. 4.10. V CI - 1.814... .
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-24-
We note that the intervals for Ct corresponding to case III
and case IV respectively overlap. The nature of this situation
will be somewhat elucidated here.
Instead of considering the pattern with which we start
in case III as made up by a net of equilateral triangles, we
think of this pattern in terms of rhombuses, with the base
angle v = 60?, corresponding to Ct = 27r/3iT = 1.209... . If
we increase v to v W 90?, that is, we change the rhombuses
There-
into squares, Ct will increase to Ct = 7r/2 = 1.571... ?
after, by "stretching" the squares into rectangles, we may
further increase Ct to Ct 3 7r/~ 1.814 ... ?
Case V.
?
2.418 ... 4' C t 167r 2.714 ...
3V 71x7'
the optimal property is possessed by a net of hexagons which
have two perpendicular axes of symmetry and can be inscribed
in a circle; in general, they are not equilateral.
We start with a net of equilateral, congruent hexagons
and place the centers of circles at the vertices, the radius
of the circles being equal to the side of the hexagons. In
this case Ct = 47r/3i3 = 2.418 ... . This case is shown in
Fig. 4.11.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Fig. 4.11. C, = 4 7 = 2.418...
We let C' increase without spoiling the property of k(p)
of having only two consecutive integers as values, by narrowing
suitably our hexagons. We can continue this procedure until
the circles corresponding to the vertices of neighboring
hexagons come in touch. Fig. 2e.12 shows the extreme situation,
which corresponds to
C' - 167/74? = 2.714...
Fig. 4.12. CI = 7 = 2.714 ...
C f
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-26-
Case VI. If
Ct _ ~ = 3.628...
the net of equilateral triangles has the optimal property. This
pattern is represented in Fig. 4.13.
Fig. 4.13. C' = ?r = 3.628 ... .
We may summarize the previoj findings as follows.
By means of lemma 4a, optimal regular nets of sample points
were found for the following values of C'
YALUE- g
C46 r = 0.907...
203
III + IV 1.209 ... 2r = C'1 ` r 1.814...
30' i 1 5'
1
V 2.418
4r < C< L611 L 2.714...
3IF "- - - 7ti7
VI C ' 3.628...
3
1
On intuitive grounds, I suspect that for case. V we should have
2.418 :.. _41 1,
(5.4) r"(u) = 4 uif 0 1.
Lemma _5a.
1 Y.n
u2
.0
(u) r" (S) du = ? a if ?> a> 0
=0 ifa> ?.
Proof. The case a > p is clear. If p. > a, we have,
in view of (5.4)
1 This paragraph is due to Hajek and Zubrzyrki.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
30
Cx+c~G J~ N.=+ kJ ~:l l7
1 rn
V
0 u
(u) r"(?) du -
(?})2 a du
r
a uV (u2 - a2)(?2 - u2)
a ?
(A) 2 - ? arc cos =
2
a
1 (A)? r 1 $
'?.7t'2a?
Theorem 5.1. The correlation functions p(u) of type
given by (5.1) are characterized by the following properties:
(d) p(u) is continuous, convex, and with p(c) - 0.
(e) p'(u) is absolutely continuous.
(f) in (1/u2) r"(a/u) p"(u) du is a nondecreasing
The functions F(a) and p"(u) are linked by the following
inversion formulas:
function of a.
(5.6) dF(a)
(5.7)
p"
(u)
r a3
13
r 00 1
0 a
d f~ 00 -lr2 r"(u) p"(u) du
~J 0 u -
r"(a) dF(a)
The condition (f) is fulfilled, for example, if(p"(u)/u) is a
nondecreasing function of u, which means, provided that p"(u) is
absolutely continuous, that
r~Y; 5
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
~J~) 3
(5.8)
p"(u) - up"(u) > 0
If (5.8) holds, then F(a) is absolutely continuous and therefore
(5.9) ?(a) 1 2 1r/2P a de
a a fO in a sing 9
Proof. The property (d) easily follows from the correr
sponding property of correlation functions r(u/a). The property
(e), and simultaneously the relation given by (5.7), will be
proved, if we show that the indefinite integral of the right
side of (5.7) equals p'(u). Now,
f ??J 00 2 r?(a) dF(a) du 00 a f?? a r"(e)d dF(a)
U 0 a 0 u
00 1 a r'(e) dF(a) pt(u
0
The change of integration order is justified, since r"(u) >_ 0
(Fubini's theorem). The last identity follows from (5.1) by
differentiation under the integral sign, which is justified,
since (1/a) r'(u/a) is uniformly bounded for u > e > 0.
In view of (5.7), we have (by using Fubini's theorem again CO 1 1 - r"(u) p"(u) du r"(u) f-4 r"(?)dF(?)du
0 =
0 u fo u 0 ?
x f~ r?? 2 r"(u) rn(y) du dF(?)
0 ? No u
which gives, in accordance with (5.5)
Approved For Release 2009/07/09: CIA-RDP80T00246A011700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
t ut WA a y ~r rw?
-32-
(5.10) J 2 ro (u) p"(u) du = f 0 - O(p)
0 u a ?
The last relation, however, is equivalent with (5.6).
Now, assume that the correlation function p(u) fulfills
the conditions (d), (e) and (f), and consider the function
F(a) given by (5.6). In view of the property (f).. F(a) will
be nondecreasing.
The fact that the total variation of F(a) equals 1 follows
from the subsequent lemmas 5b and 5c and from the following
relations:
(5.11)
00 1 - $ a3 J 2 r" (u) p" (u)du +
0 u 0
+ . f r(u)da p"(u) du =
fo[ 0
$
u
/' up" (u) du = - ~ mp v (u) du =
~f 0 0
= p(0) p(oo) = 1.
It remains to show that p1(u) is uniquely determined by the
relation (5.6), that is, that p(u) coincides with the correlation
function obtained from (5.1). However, (5.6) is equivalent to
(5.10), if rewritten in the following four:
r"(1) d o!(u).
(5.12) $ r0 3 dF(?) J0u-
JJ 0 ?
dF(a) a* a3 d f - r"(u) p"(u) du da =
I
0 0 0 u
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Comparing (5.12) with (5.7), we can see that the p-(u) may be
determined from (8/Tr) f ??(1/?3)dF(p) in the same way as F(a)
a
has been determined from pit (u). , (The fact that p* (u) may not
haves finite.. variation is irrelevant.)
Before proceeding to the rest of the proof, we observe
that by substituting u = (a/sin 9) into (5.6) we get
r/2 4a
(5.13) dF(a) a3 d f an e
0 sine
d 6
From this form it is easily seen that the condition (f) is
fulfilled if (p"(u))/u is a nondecreasing function of u, or,
more especially, if (5.8) holds (notice that L(p"(u))/u]'=
1/u2 (p"(u) - up"' (u)j). Now, if (5.8) holds, we can
differentiate in (5.13) under the integral sign (FuIinits
theorem), which gives
Tr/2
.14) dF a 1 Yo a 1' a sin 9
(5 =
a P (sin g)~ P s n 9) a2 da
Tr/2 Tr
a 2
- aP?(s"' ?)cos
a2f0 pill (s a 8)de +
ill
. e
f 2 ,,, a cos` 0 d -
2 aP (sn6) sin20
r/'2 t3_) $i2 a f. cost
2
0 P'ti S'..n ;iil`` H
f
_ 1 2 I 7T./2 el '
2 p $ 11 A
0 S:n b
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
Fuji
The relation p"(co) = 0 which we have used, follows from the
subsequent lemma 5c. Our theorem is thus completely proved.
Lemma 5b.
(5.15)
f 00
0
u2 r" (u) du = 8
3r
Proof. Integrating by part, we have
f fo 0 u2 rot (u) du - 2 u r i (u) du
-
0
u 1 - u2 _ J. ?
37
Lemma 5c. Any correlation function p(u), which fulfills
the conditions (d), (e) and (f) of Theorem 5.1, has the
following properties:
(5.16) lim a3JO 40
a 0
1 r"(u)p"(u) du =
U2
00
= Jim a3f . r"(u)p"(u)du = 0.
a-. o0 0 u -
(5.17)` Jim a pf(a) = Jim a p1(a) = 0.
a-* 0 a-~ co
If, moreover,(p"(u))/u is nondecreasing, then
(5.18) Jim a2 p"(o) Jim a2 p"(a) = 0.
a-~ 0 _ a- - .
Proof. As p(u) is convex, - pt is nonnegative and
nonincreasing, and we have
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
101
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
a ~ 1 2
= J f r11(u) p"(u)du d? =
u
a/2 0
a
0 < - 2 a p+(a) _ - j pt(u)du = P( ) - P(a).
a/2 -
As p(u) is continuous in points 0 and co, (5.17) is clear.
Now, if (5.8) holds true, the function (p"(u))/u is
nonnegative and nonincreasing,:so that
(5.19) Oa3a2P,,(a) a 1 3a3 "(aa ) =a2 f a (uu) dus
a/2
ra
tar p"(u)du = 2a p+(a) - p'(2 a)
J3/2
where the expression converges to 0 if a ---> 0, or a --a ~.
The same consideration will be used in proving (5.16).
In view of the condition (f), we have
2 rtr(u)p" (u)du =
fo (5.20) 0 < 3 a3 Go
u
a ? 2
J~?l?LS
JO'O
a/2 u
-35-
r"
r
tj
u)du =
a/u 2
u r"(u)dq up"(u)du
[au
where, in view of (5.15)
j a /u 2 " u r (u)du
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: C} IA-RDP80/T002Q46AO11700340001-4
Consequently, on the one hand,
(5.21) rO if a/u u2r"(u)du up"(u)du <
J
-a/2U
and, on the other hand,
a /u
(5.22) fo Lf8i u2r"(u)d up"(u)du'=
37
lc~
a/e
up"(u)du
0
CO up"(u)du.
37 a
The inequalities (5.20) together with the inequality (5.21)
or (5.22) prove the relation (5.16) for a> 0 or a
respectively. (Notice that up"(u) is integrable with
fup"(u)du = 1.)
Ianple 5.1. The convex correlation function a-cu has
a negative third derivative and therefore fulfills the condition
(5.8). Hence, it admits the representation (5.1), where the
spectral density is iven (5-9).
Example 5.2. The convex correlation function
u
P(u) _ _ r2-_--, [ du
also admits the representation (5.1), since (p"(u))/u =
e-(u2/2) is a nonincreasing function of u.
ExamPle 5.3. The linear'convex correlation function
p(n) 1-u ifu1l
=0 if u> 1
has a discontinuous first derivative, and therefore, does not
dmit the representation (5.1). It may be shown, however,
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
FUR bb uS j 3a ?ef/ 7.'~ r' V
that p(u)
Actually,
we have
is not a planner isotropic correlation function.
considering a square net of points with coordinates
0 i,
j = n - 1
E E p(IPij - Pi' j, I)(- i+j+iI+jt
i,j=0 i+,jI=O
n2 - 4(1 - 2 ) n(n - 1)
which is negative for .a sufficiently large n.
Example 5.3 shows that the class of planner isotropic
convex correlation functions is smaller than the class of
linear convex correlation functions. One might suspect that
all planar isotropic correlation functions are expressible
in the form (5.1). This is, however, disproved by the following:
Theorem 5.2. There exist isotropic stationary
stochastic processes in a plane with correlation function
g(x,y) = f(\1 x2 + y2 ) such that f(u), 0 4 u is a convex
function which cannot be represented in the form
(5.23)
where
(5.2k)
f(u) = f r(~) dF(a),
0
r(u) = .(arc cosu-uYl-u2) if 01u11,
0 if 11u
and F(a) is a distribution function with F(0+) = 0.
EI. ,I L' i
u~%iuvy~!
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
This theorem follows from the following two lemmas:
Len,_n 5d. The function g (x, y) - f ('x2 + y2 ' ) , where
(5.25) f(u)
22r ((arc sin u)
-2r(W-u)
for 0!u
(u - u - 1 ) )
for 11uIw,
is a correlation function of a stationary isotrovic stochasticc
process (see Fig. 5.1),
IV T
/
1 2 3
Fig. 5.1. V = f(u), f(u) given by (5.25).
Le=se 5e. If the function f(u) is representable in the
form (5.23), then f"(u) > 0 for all u > 0 with u < a, where
F(a) < 1.
To prove lemma 5d, we consider a linear stationary
stochastic process t(t) with correlation function h(t) given by
(5.26) h(t) - 1 - It) if Itl =.10
- 0 otherwise.
Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
 Approved For Release 2009/07/09: CIA-RDP80T00246AO11700340001-4
-39-
Define then a, plane stochastic process t(x,y) by
putting
(5.27) =.r(x cos a, y sin a),
where a is a random variable independent of the process
r(t) with
0 if a 0,
(5.28) Pr (a a)= a/2ir if 0 a 2r,
1 if 2ir e a .
In other words, we first define a plane stochastic process
which depends only upon one coordinate and has with respect
to it correlation function h(t), and than we randomize the
direction. It is aeon that t(x,y) is an isotropic stationary
stochastic process with correlation function g(x,y)
f( x2 + y2 ), where
(5.29) f(u)
1 r 27
2v f o (1 - h(u sin fir)) d*
which leads to (5.25).
To prove lemma 5e, we note that the second derivative
of a function f(u) given by (5.23) is given by
(530) fo(u) a J O e r" (M-) dF
where
r" (u) S 0, if 0 < u < 1,
41 - u2
=0 if l