О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики
Настоящая статья состоит из двух частей. Хотя задачи, рассматриваемые в них, на первый взгляд, кажутся мало связанными, в действительности же их объединяет общность метода исследования и внутреннее единство. В первой части доказываются некоторые теоремы, относящиеся к условному распределению функц...
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| Date: | 1953 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1953
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7755 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512745297805312 |
|---|---|
| author | Гихман, И. И. Гихман, И. И. |
| author_facet | Гихман, И. И. Гихман, И. И. |
| author_sort | Гихман, И. И. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:11:31Z |
| description | Настоящая статья состоит из двух частей. Хотя задачи, рассматриваемые в них, на первый взгляд, кажутся мало связанными, в действительности же их объединяет общность метода исследования и внутреннее единство. В первой части доказываются некоторые теоремы, относящиеся к условному распределению функционалов от последовательности независимых случайных величин, во второй — рассматриваются вопросы, связанные с распределением колмогоровского критерия согласия в том случае, когда проверяемая функция распределения содержит параметры, определяемые эмпирическим путем. |
| first_indexed | 2026-03-24T03:33:40Z |
| format | Article |
| fulltext |
1953 YKPAHHCKHH MATEMAT~4ECKHH~YPHAfl
HHCTHTYT MATEMATHKH
T. V, N2 4
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saeMbie B HHX, Ha rrepBbiH B3fJI5I.U, Ka)KyTCH MaJIO CB5I3aHHb!MH, B ,n:eiicTBH
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e,n:HHCTBO. B rrepBOH 'IaCTH .UOKa3b!Ba!OTC5I HeKOTOpb!e TeopeMbi, OTHOCHW:HOCH
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k vn IJ7/nk= ..E~ •.
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(5)
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415
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l n
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o, (1)
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n
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h(s-np)
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00
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-oo
p. = P{~~: =a+ sh}.
ITyCTb
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•
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2n
r ,n:e OIJ.eHKa 0 ( l) paBHOMepHa OTHOCHTeJib~O Z.ns. - 00 < Zns < 00.
(8)
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1 n-1
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Y n 1
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(9)
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{ } p,P/ (s-r)
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d'(e, n, Zm) = .2) P {x,,fz,..},
I ""r I> •
a 1 (e, n, Zn8 ) = .2) X11 r p {xnr/Zn,},
I "nr I< •
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I Xnr I < •
Hs (8) ,li.JIH Pn' (s- r) CJie,ll.yeT
(zns-xnrl~ -- , h - __ 2 _ _ _
Vn-1 Pn (s- r) = v- e + o(l).
2n
(10)
(11)
(12)
(13)
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• Xnr - - +x z
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6hiX 3HaqeHH5IX Xnr
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--
TO
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I "nr I > • E I "nr I > • n
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npH JIJ06oM cpHKCHpoBaHHOM e > 0.
417
•
al(s, n, z,..) = I Xnre- x;,.+xn,.zn8p,(l+o(l)) =
I Xnr I < '
AHaJrorutJHO noJiytJHM
1 o, (l) O(e)
(~~(s, n, z,..) =- + --- + -- .
· n n n
Oepeif,n:eM K .ll:OK333TeJibCTBY TeopeMbl 1. ,UJIH orrpe,n:eJJeHIJOCTH OITHTb
npe,n:noJio:tKHM, tJTO /;1,- perneTtJaTbie seJIHtJHHbi. Ope:tK,n:e scero 3aMeTHM,
'tJTO eCJIH cp·UKCllpOB3Tb BeJilltJllHY "'n = "J.nn• ITOJIQ)!{lfB
( 15)
TO nocJie,n:oBaTeJibHOCTb "!no, "lnh ... , rJnn o6p::~3yeT u.errb .MapKona. T aKHM
o6pa30M, 3a,n:aqa orrpe,n:eJieHHH rrpe,n:eJibHOH yCJIOBHOH BepOHTHOCTH C06b!
THH (5) rrpH rHITOTe3e ( 15) MO:tKeT 6b!Tb perneHa MeTOJl:3MH, pa3pa60T3HHb!MH
,n:JIH u.eneH: MapKosa [ 10, 11].
Ope,n:noJIO:tKHM, tJTO 3a,n:aHo "'nk =Sn, "'nn = z,.. 0oJib3YHCb JieMMOH 1,
" . sk+l
noJiytJHM ,n:JIH ,ype3aHHbix yCJioBHbix MoMeHTos seJIHtJHHbi 7Jnk+l -1Jn1: = Vn
CJie,n:yiOlli.He ou.eHKll:
:r,n:e
0 (1)
1-a. (s, n, S,., Zn) = -- ,
n
1
a2(e, n, Sn, z,.) = (1 + o.(l) + O(e)) - ,
n
k
t.,.= - <.T < 1. n
00Jib3YHCb npe,n:eJibHb!MH TeopeMaMH .ll:JIH ITOCJJe.II.OBaTeJibHOCTeif CJiyqaif-
HbiX seJIHtJHH, CBH3aHHbiX B u.e rrb MapKosa (eM., HarrpuMep, [11]) u rrpHHHMaH
'BO BHHM3HRe TOJibKO-tJTO UOJiytJeHHb!e COOTHOllleHUH, Herrocpe,n:CTBeHHO ITOJiy
'HlM Jl:OK33biB3eMOe.
H3 ,n:oKa3aHHOH TeopeMbi JierKo
BMeCTHYIO cpyHKIJ.HIO pacrrpe,n:eJieHHH
rrpu rrrnoTe3e "'n = Zn(n __.,. oo, Z11 -z).
rroJiytJHTb rrp e,n:eJibHYIO YCJIOBHYIO co
napbl cpyHKU.HOH3JIOB {mn (t), Mn (t)}
0Ha pasHa [11, 12]
b'!.r
1 00 '
f1J(a, b; t, z> = v- I{exp { -2r(b+a) [r(b+a)-zJ }fe-u du-
2n - oo
b2r-t
-exp{-2(rb+l=la)(rb+r-la-z)} J e-u'du}, (16)
-a-tz-2r(l -t) (b+a)
a,.= Vt)-t) ;
b-tz-'2r(1-t) (b+a)
b - . .
2r- Yt(l-t) '
(17)
-a-tz-2 (1-t) (rb+r-Ja) a - .
2r-1 - vi (1 - f) '
b-tz-2(1-t) (rb+r-1a)
b2r-1 = Vf(l-f) •
ilpe.n:eJibHOe YCJIOBHOe pacrrpe.n:eJJCHHC BeJJH'lHH
1 ' k k n '
M'(n, t)= max v-1 ~sr--I~·)·
O~k< nt n \ 1 n 1
m' (n, t) =- min ---= ~' §r-- _Is. , J(k k" )'
O<k<nt V n 1 n 1
1 n
npH mnoTe3e --= Isr=zn(zn-z) pasHo u(O, 0), r.u.e u(t, s) ecTh pe
V n 1
llieHHe ypaBHeHHH (6), Y.Ll:OBJieTBOpHIOlli.Ce rpaHH'lHb!M YCJIOBHHM
u("J", -a+-r;z)=u(c, b+rz)=O, 0 <;::: -r; <t< 1,
u(t,s)=1, -a+tz < s < b+tz.
no.n:CTaHOBKa y = s + tz rrpeo6pa 3y CT ypaBHCHHe (6) K TOMY Me BH)J.Y,
r.n:e TOJibKO HYR<Ho noJIO:lKHTb z = 0, a rpaHH'lHbie ycJJOBHH 6y.n:yT
u("J", -a)=u('r, b)=O
u(t, s)=O
0 <;::: -r; <;::: t < 1,
-a < s<b.
TaKHM o6pa3oM, HCKOMoe ycnoBiwe pacrrpe.n:eJI.eHHe oKa3biBaeTCH He3asu
CSI!l.{HM OT Z .H p-aBHO ai (a, b, t, 0).
no CHX rrop Mbl C'lHTaJJH t < l. Ope.n:eJibHbiC pacrrpe.n:eJJeHHSI .l{JISI CJiy
'laSI f = l MO:lKHO ITOJJY'lHTb HCITOJ!b3YSI TeopeMy pa60Tbl [JJ]. B CHJJY :noli
TcopeMbi
W(a, b; 1, z) =lim 0(a, b; t, z).
1-+1
Teo p eM a 2. EcJJH rroCJie.n:osaTeJibHOCTh cJiyqaifHhiX neJIH'lHH {§n}
y.n:osJieTBOpSieT ycJIOBHSIM TeopeMbl l, Zn --· B cnyqae a) o.n:Ho 'H3 B03MO:lKHhTX
3Ha'leHHH CYMMbl
1 :., J:
71n = v- .L. "'"' n 1
TO I) yCJionHaSI <j:lyHKll,HSI pacrrpe.n:eJieHHSI rrapb! BeJJH'lHH
llpH rHllOTC3C TJn = Zn CTpCMHTCSI K
Ql)
- . 1 )\'I: m•nv- ....... "'"· lh k < n n 1
_2.' exp{ -2r(a+b)[r(a+b)-z]} -exp{ -2(rb+r-la) (rb+r--Ia-z}; -·
419
2) rrpe,l!,CJibHaH ycJIOBHa5I ¢YHKUHH pacrrpe,l!,CJICIU!51 BCJIH'IHH
1 ( k k n 1:) _ . 1 ( k k n 1:) max y-- ,I g.- - .2) ,. , mm 11- .2; s, - - .2; "• '
o~k< n n 1 n 1 O<k~n I Jl 1 Jl 1
rrpu rurroTe3e "'n = Zn He 3aBHCUT OT z u paBHa
00
1 + 2)2 exp { -2r(a+b)1}-exp{ -2 (rb+r-1a)~}-exp { -2(r-1b+ra)2}.
1
3aMeTHM elll,e, qTQ B CHJIY p e3yJibT3TOB y:>Ke UHTHpOBa~IHOH pa60Tbl (11) B TC
opeMe 1 MO:>KHO paccMaTpHBaTb H ueorpauuqeHHbie peryJIHpHbie o6JiaCTH,
B qacTHOCTH MO:>KHO 0606Il.LHTb TeopeMy 1 H Ha 06J13CTH BH,IJ,a
{ -a.<TJnltr<b.; r=1,2, ... ,m }• (18}
r,IJ,e k.- t., 0 < t. < 1. BrrpoqeM, rrocJie,rumil pe3yJibTaT MO:>KeT 6hiTb noJiy-
n
qeH H HCITOCpe,.a:CTBCHHO C ITOMOII.I:b~ JIOK3JlbHbiX rrpCJlCJibHbiX TCOpeM )lJIH CYMM
HC33BHCHMb!X CJiaraeMbiX. Ilpe)lCJibHOe COBMCCTHOC yCJIOBHOC pacrrpe,.a:eJICHHC
BeJIHqHH TJ,, k/ ~-f., 0 <f.< 1, rrpH rHITOTC3e 7Jnn = Zm Zn --:> Z, COBITa,naeT
C .COBMCCTHbiM pacrrpe,llCJlCHHCI\1 BCJIHliHH
{?J{ti) + tiz, '7(t2) + t2z, ... , 7J(t,,.) + tmz},
r,IJ,e 1J (t)- cJiyqaH:naH ¢YHKUH51 HerrpepbiBHOro MapKoBcKoro rrpouecca, ynpaB
JI5IeMoro JlH¢¢epCHIJ.HaJibHbiM ypaBHCtiHCM
iJu X OU 1 01U
ot -1-t ax+ 2 £ix1 =O. (l 9)
CJiyqaifHbiH rrpouecc {rJ (t) } 6biJI ITOJlp06Ho paCCMOTpeH ny6oM [13] B CBH3H
c 3BpHCTHqecKHM o6ocnoBaHHeM TeopeM A. H. K:oJIMoropoBa [7] ·H H. B. CMHp
HOBa [6). 3TOT rrpouecc ITOJIHOCTbiO xapaKTCpH3YCTC51 CJICJlYIOlli,HMH CBOHCTBa
MH: a) m-MepHaH ¢YHKUH5I pacrrpe,lleJieHH51 BeJIHqHH 1] (ti), 7J (f2), ... , 1J (tmJ
HOpMaJibHa, 6) M17 (t2) 7J (ti) = ti ( 1 - t2), ti < t2 < 1. OT cxOJlHMOCTH ycJioB
noti BCp05ITHOCTH Ha UHJIHHJlpHqecKHX MHO:>KCCTB3X JierKO MO:>KHO nepeHTH
K CXO,ll;HMOCTH ITO Bep05ITHOCTJ:I BCCbMa 06lli,HX ¢YHKUHOHaJIOB OT CJiyqaiiHb!X
<PYHKUHH. Mb! paccMoTpHM HH:>Ke OJlHH rrpuMep TaKoro po,IJ,a, noCJiy:>KHBIIIHfl
npe,llMeTOM CTieuHaJibHOrO HCCJIC)l0B3HHH (4}
IlyCTb f (x) HenpepbiBH351 ¢YHKUH5I, onpe,lleJieHH351 Ha (-<X>, <X>), Y.ll.O'B
JICTBOp5110lll,a51 ycJIOBHJO JlumuuTua
Onpe,IJ,eJIHM
I !(xi)- f(x2)1 < C I XI- x2l·
I= f f [?J(t)] dt
0
KaK npe,lleJI B CMbiCJie CXO)lHMOCTH B cpe,llHeM no TIOJiyyrropHJlOqeHHOMy MHO
:>KeCTBY pa36HeHUH cerMeHTa [0, 1] HHTerpaJibHb!X CYMM
n
.2; I [ 11 (t,')] Llt,
i=l
Llt1=t,-f,_1 , t0 =0, t,.=l, (-1-<.t,'-<.t,,
eCJIH TOJibKO 3TOT rrpe.neJI cymecrByeT.
(20)
BnpoqeM, cymecTBOBaHHe 3Toro mnerpaJia .ZI.JIH cpyHKU.HH, y.n.oaJieTBO
pHIOW.Hx ycJIOBHIO JlHnWHTU.a, ycTaHaBJIHBaeTCH 6e3 Tpy.n.a. ECJIH { L1ti, k} -
HeKOTOpOe IlO.ZI.pa36HeHHe pa36HeHH51 { LJf; }, TO
n i..: i
-< C _1: 2,.M I'tJ(t,')--1J(li,k)I'L1ti,k·
i= l k=l
C .n.pyroii: CTOpOHbl,
M l11(t2)- 17(ti) [ 2 = (t2- t1 ) (1- tz- t1), fz > t1.
0TCIO.ZI.a CJie.n.yeT
<5 <; Ct, (20')
r.ne e =max L1t;. TaKHM o6pa3oM, HHTerpaJihHbie cyMMhi (20) npH e ~ 0
l ··~ l < n
-cxo,nHTCH B CMhiCJie cxo.n.HMOCTH B cpe.n.HeM. DoJIO:tKHM
(21)
H pacCMOTpHM cyMMbl
I:= ~ k:: f [ 1] ( ~) ] ,
JI eM M a 2. ECJIH f (x) y.nonJieTBOpHeT ycJIOBHIO JlHilW'HTU.a H BbiilOJIHeHbi
npe.nnoJio:tKeHHH TeopeMhi 1, TO ycJIOBHoe pacnpe.n.eJieHHe cyMMhi In npH mno
re3e "'n = Zn CJia6o CXO.ZI.HTCH K pacnpe.n.e.T!emno cpyHKU.HOHaJia I.
Bne.n.eM xapaKTepHCTHqecKHe cpyHKU. IiH 1/ ' (i.), 1.< (1.), tfln, m (i.) BeJIHqHH
I, I:, In, m. COOTBeTCTBeHHO. Tor.na w~P-) ~ tf'U-) npH n- <Xl paBHOMepno
OTHOCHTeJihHO }. n JII060l\r KOHeqimM HHTepnaJie 12 1 <A. DycTb n- cpHKCH
poBano. 11MeeM
lt/J(i.)-t/•,.m (i.)l<llf;(i.)-w:,(l.)l +I t/J:(i.)-tf.'n,m(i.)l + 11/Jn m(A)-t/Jnm,t(A)I.
llJI5I JIIo6oro cpHKCHpoBamwro ~-:> O no.n6ep.eM n raKHM, qro6bi I ttt(A)- tp~ (!..)I <
13 < 3 , I i.l < A. Tor.na npH m ~ =
t/J: (i.)- 1/Jn, m (A}- 0.
PacCMOTpHM Tenepb pa3HOCTb tfln, m ().) -1/-J,m, 1 (i. ). 11MeeM
I tf.',, m (A) -1/Jnm, 1 (A) I -< I), I M {g/'111 m = Znm },
r)le
g=mm 2, - - :E f (~km)-- 2) f (~i) · . { 11 " 1 nm I}
n k=t nnz k=I
• .P11 • 111 {J.)- ycJIOBHan xapaKTepHCTH'ICCKan ¢YHK!I,H51 BCJIH'IHHbl In. , , coOTBeTcTny
mlll,aH runoTe3e '1nm = z11111•
421
TaK KaK
H
I Skm -Sm{k-IJ+r i<G!~_•J + ~- ~-- ~ I ~·I•
n Vnm s=(k~l)m+r
. 1 k .
TO B CHJIY JieMMhi I (. rrpnMer-rmr ee K rrocJie,n:osaTeJihHOCTH §~:' = y- J; Sr_)
m (k-l)m+l .
Mbl IlOJiyqnM
M{ I = 1 . ./' _!j_ n m { ~,,J 0 ( j _ ) l - 0 ( __!__ )
U,1Jnm Zm,., "':::- nm ~ ~ n + y; J- Vn ·
TaKHM o6pa30M, paBHOMepHO OTHOCHTeJibHO m pa3HOCTb 'lfln, m(l)-'!Jinm(j.)
cTpeMHTCH K 0 rrpH n ---:> oo. H3 3Toro CJie,n:yeT
I t/• {).)- lf1nrn ().) I < E
;J:JI5I BCeX p:OCT3TOqHo 60JibillHX tn > m 0 (e) rrpH HeKOTOpOM <lJHKCHpOBaHHOM
n = n(e). Ho .'!erKo 3aMeTI:ITh, qro
11./JN()..) -'lfnm (i..) I-+ 0,
r,n:e m= [~]- ~eJiaH qacTh OTHorneHHH ~. CorrocraB.TJHH 3To c rrpe,n:br,n:yll.J,HM
HepaBeHCTBOM, IlOJiyqnM ;J:OK33bJBaeMoe.
JleMMy, ,n:oKa3aHHYIO ,n:JI5! ¢YHK~HH f (x), y,n:osJiersopmomnx yCJioBHIO
JlH!lillHT~a, JierKO Terrepb o6o6ll.J,HTb Ha 6o.'!ee 06ll.J,HH KJiaCC ¢YHK~HH.
DycTh {fj." (x)}, {t; (x)} - ,n:se li.WHOTOHHhie rroCJie,n:osaTeJihHOCTH ¢YHK
~Hi'l, rrpHqeM rrepBa5I- MOHOTOHHO y6bJB310lll,35I, BTOpa5I - MOHOTOHHO B03-
paCT310lll,35I, cxo,n:HmnecH K HeKOTopoii ¢YHKU.HH f,., (x). DoJIO)!{HM I+ (N) =
= I(fj"), I-(N) =I(!; ), H aHaJiornqHo I/;(N), r;:(N). Tor,n:a, eCJIH ,n:mi
¢YHK~HH /;J, r;;· YTBep)!{,n:eHHe JieMMbl HMeeT MeCTO, TO
It(N) > In ( './) > r ;: (N), I-(N) <lim In( oo) < lim In (oo)<: I+ (N).
DoKa)!{eM rerreph, qro rrpH N---:> oo I + (N) - I- (N) cxo,n:nrcH B cpe,n:HeM
K HYJIIO. H3 3Toro 6y,n:er cJie,n:osaTb, qTo rroc.ne,n:osaTeJihHOCTH HHTerpaJioB
{ I+(N) }, {I-(N)} rrpH N---:> oo cxo,n:HTCH B cpe,n:HeM K o,n:HoMy .n TOMY )!{e
rrpe,n:eJiy, KOTOpb!H MO)!{HO orrpe,n:eJIHTb K3K
I(f) = f f[1J(l)] dt,
0
rrpnqeM lim In (f) TaK/Ke cymecTsyer H rroqTH Hasepuoe cosrra,n:aeT c I (f).
Ml I+ (N)- z- (N) I' < f Ml fit ['I (t)] -- r;; [1,' (t)] I' dt.
0
TaK KaK pacrrpe,n:eJie~me 7J (t) rrpH ¢nKCHpOBaHHOM t a6comon10 HerrpephiBHo
OTHOCHTeJibHO Jie6erOBOH Mepb! Ha rrpHMOH, TO H3 CXO,UHMOCTH { 1t(x)-r;; (X}'
K HYJIIO IIOliTH BCIO.UY ua IIp5Il\WH CJre,n:yeT TaK)KC CXO,UHMC>CTb f ;:t· [1) (f)]
- r;- r '1 (t)] K HYJIIO rrpu N ~ 00 IIOliTH HaBepuoe. DpHMeHHH T.eopeMy Jle-
6era o rrepexo,n:e K rrpe,n:.eJiy rro,n: 3HaKoM HHTerpaJia, rroJiyliHM Tpe6yeMoe;
Mbr ,n:oKa3aJiu cJie,n:yromyro JIC\-tl\ry.
Jl eM M a 3. DycTh f (x)- rrpoH3BOJihiiaH orpaHHliennaH H H3MepuMa5I
rro Jle6ery QJ YHKU:HH; Sni.: onpe,TJ:CJIHeTCH cooTnorneHHHMH (21), H BCJIHliH
Hbi !;k y,n:oBJi eTnopmoT yCJioBHHM TeopeMbi 1.
Tor,n:a ycJIOBHoe pacrrpe,n:eJieHHe cyMMbi
rrpu rurroTe3e
1 " -- I f[S", k]
n k=l
1 ,:. - :: ---= ~ (~k M,k)- Zn,
~ n k= t
r,n:e Zn- ,UOIIYCTHMOe (B CJiyliae peme TIJaTbiX BeJIHliHH) 3H3lieHHe CYMMbi
CJieBa, H Z 11 ~ Z IIpH 1l - 00 CJia6o CXO,UHTCH K pacnpe,n:eJICHHIO QJYHKU:HOHa~Ia
I (f) = I f[r; (f) l dt,
0
r,n:e TJ (t)- CJiytiaiinaH cpyHKU:HH MapKOBCKOro rrpou:ecca, yrrpaBJIHeMoro ypaBr
HeHHeM ( 19).
3HatieHHe 3TOH JieMMbl COCTOHT eme H B TOM, liTO OHa CO,n:ep)KHT B ce6e
HI!B3pHaHTHOCTb rrpe,n:eJibHOfO pacrrpe,n:eJieHI15I IIO OTHOIJJCHHIO K ,UOIIYCTHMbiM
pacrrpe,n:eJI,eHHHM aeJIHliHH !;k. TaK, narrpHMep, ecJIH npHHHTh, liTO
f(x)={ol x>O
x <, O,
CJiyliaHHbie BeJIHliHHbi f;k = + 1 C O,UHH3KOBOH Bep05ITHOCTbiO H Z = 0 (pac
CMaTpHBaeMOe HaMU pacrrpe,n:eJieHHe He 3 3 BHCHT OT z), TO MO:ii<HO BOCIIOJib-
30B3TbC5I y)Ke H3BeCTHbiMH pe3yJihTaTaMH o6 yCJioBHOM pacrrpe,n:eJieHHH IJHCJia
k
IIOJIO)KHTeJibHbiX CYMM <rk= _}; gr IIpH fHIIOTC3e a11 = 0 [1, 2). 3TO IIpHBO,UHT
1
K CJie,n:yrorn.eii TeopeMe.
Teo p eM a 3. Dycrh Y 11 o6o3Hatia~~T liHCJIO IIOJIO:i!UITeJihHbiX tiJieHOB
J ~~ 1: - !!_ ~ E I c B IIOCJIC,UOBaTCJ!bl!OCTH li"" ,r n -f' .r1. H Sk y,n:OBJICTB0p5I!OT YCJIOBHHM Te-
opeMbl 1. Tor,n:a rrpH n ~ oo yCJioanoe pacnpe,n:eJienue OTHOrneHHH
n
(22)
1 n "
rrpu runoTe3e -- I~r=Z11 , Zn ~ z CJia6o cxo,n:HTCH K paBHOMepnoMy pacnpe
IZ 1
,UeJieHHIO.
l.JaCTHb!H CJiyliaii TeopeMbl 3, COOTBeTCTBYIOlli.HH CJiytiaiO Z = 0 H BeJIH-
1IIUiaM f; k> IIpH!IHMaiOW:HM 3 HatieHH5I , KpaTHb!e 1, 6biJI Ue,UaBHO paCC!110rp eH
B y:tKe ynOMHHYTOH p a6oTe [ 4).
TaK KaK rrpe,n:eJrhnoe ycJioauoe pacnpe,n:eJienHe He 33BHCHT OT z, TO 6e3-
ycJIOBHoe pacnpe,n:eJieriHe OTHOWCHHH (22) IIpH n ~ oo T3K)KC CXO,UHTCH
K paBHOMepHOMy pacnpe,n:eJi eHHIO.
423~
II
Dpe,a:JIO)!<eHHbiH A. H. KoJII\wropoBbiM B 1933 r. [7] KpHTepHif corJiaCHH
Me)!{,a:y 3MnHpHlfeCKHMH )J:3HHbiMH H 33)J:aHHOH (,TeopeTHlfeCKOH") <iJYHKUHeH
pacnpe,a:eJieHHH F(x)
K_v= max JFN(x)-F(x)J,
-:o <x<oo
(1)
r.n.e F N(x) - 3MnHpHlfeCKaH <PYHKUHH pacnpe_ueJieH"HH, nocTpoeHHaH no N
pe3yJibT3T3M H36JIIO,ll.eHHH, 5JBJI5IeTCH, I<3K H3Be CTHO, YHHBepC3JibHbiM B TOM
·cMbiCJie, lfTO 33KOH pacnpe.n.eJieHH51 BeJIHlfHHbi K Jl.- He 3aBHCHT oT n e npepbiB
HOH <PYHKUHH F (x). Dpe.n.eJibHbiH 33KOH pacnpe,a:eJieHHH BeJIHlfHHbi
K N (N ~ oo) Ta6yJinpoBaH [14], HMeioTcH TaiOKe Ta6JIHUbi pacnpe,a:eJieHHH K N
,li.JIH KOHelfHbiX N [15). Bo MHOrHX CJIYlf35IX, O)J:H3KO, OTHOCHTeJibHO <!JYHKUHH
F (x) H3BeCTHO TOJibKO, lfTO oHa npnHa,li.Jie)KHT K HeKoTopoMy Tnny pacnpe
)J,eJieHHH, 33BHCHIUHX OT napaMeTpoB 0, F (x) = F (x, 0), np·nlfeM 3HalfeHHH
noCJie,a:HHX onpe.n.eJIHIOTCH 3MnHpHlfeCKH. B 3TOM cJiyqae BMeCTO BeJIHlfHI-Ibi ( l )
npHXO)J:HTCH BBO)J:HTb BeJIHlfHHY
K N = max I FN(x)- F(x, ii) J,
-eo<.r
(2)
f.ll:e 0 - 3MDHpHlf0CKOe 3H3lfeHUe napaMeTpa 8, H DOJIO)!{·e~JHe pe3KO MeHHeTC51.
IloCJie)J,Hee CBH3aHo c TeM, lfTO BeJIHlfHHa KN npn N -~ oo HeycTOHlfHBa OTHO··
CHTeJibHO napaMeTpa fl H B TOM CJiyqae, KOr,a:a norp,elllHOCTb B onpe.n.eJieHHH
napaMeTpa .cTpeMHTCH K 0 BMecTe c N-1. 0)J,HOBpeMeHHO c 3THM BeJIHlfHHa
kN nepecTaeT 6biTb YHHBepcaJibHOH, )J,a.ll<e ecJIH MeTo,a: oueHKH napaMeTpa
CTaH,ll.3pTH30B3H. l13 pe3yJibT3TOB H3CT05IIUeH CT3Tb'H CJie,J.yeT, lfTO B 3TOM Ha
npaBJieHHH npocrbie 33KOHoMepHOCTH OTCYTCTBYIOT.
Pa306beM npHMYIO - oo <X< oo Ha n HHTepBaJIOB, Ha30BeM HX HH
T,epBaJI3MH rpyrmnpoBKH, H nycTb F;(O) - Bepo5ITHOCTb cJiyqaifHoii BeJIHlfl!He
npHHHTb 3HalfeHHe, JJe)!{amee B nepBbiX i mnepBaJiax, a F,- 3MnHpHlfecKoe
pacnpe.u,eJieH'He, T. e. OTHOllleHHe lfHCJia Ha6JIIO,UeHHH, nonaBlllHX B nepBbie i
HHTepBaJIOB rpynnnpoBKH, K o6meMy lfHCJIY N npoH3Be,a:eHHbiX Ha6JIIO)J,eHHH.
)laJiee, nycTb HCTHHHoe 3HalfeHue napaMe Tpa paBHO 00 , a Pi- o6o3HalfaeT
Bep05ITHOCTb CJiyqaihiOH BeJIHlfHHe npHHHTb 3H3lJeHJie , JJe )K3IUee B i-TOM HH-
TepBaJI.e rpynnnpoBKH npH 3HalJeHHH napaMeTpa (J=80 • DoJIO)!{HM
~= VN{F,- F,(fi) },
~= Y N { F,- F, (Oe)- (8-a0 ;
0~~:o))},
,,=~-~, = V N { F, (H) - F, (Oo)- (o-ol.; 0~~:o )) },
(3)
(4)
(5)
f)J,e (01 ; fi~) 0603H3lfaeT CK3JI5IpHOe npOH3Be,a:eHHe ,a:Byx m-MepHbiX B eJIHlfHH,
i:JF, (fJo)
a --·-- - rpa,a:HeHT cpyHKUHH F, (fi) no nepe Me iHIOH o, B351TbiH B TOlfKe 00 •
oOo
Hawa 33)J:3lfa COCTOHT B HCCJie)J:OB3HHH rrpe,UeJibi!OfO pacrrpe,a:eJieHHH
max I l;i I npH o,a:noBpe Me HHOM BbinOJIHeHHH coonioweHHH N-~ oo, max P; ~ 0.
O< i .;;n i
YcJioBHMCH Ha3biBaTh .nse rrocJie,nosaTeJihHOCTH cep11i1 c.'IY'IaHHbiX se
JI 114111-1
y=l, 2, ...
(6)
(7)
3CiiMITTOTI14CCKH 3KBHB3JletiTI-IbiMI1, CCJIH Ilj)H J' ~ CXl rrp e,nCJib!-lb!C COBMCCT!-Ib!C
pacnpe,neJICHH51 M3KCHM3JlbHOrO H MHHI1M3 JibHOrO 4JI CHOB B cepHHX (6) H (7)
CYLL~CCTBYIOT O).l.HOBpeMCHHO, 11 ,eCJII1 cyLUCCTBYIOT, TO COBna,naiOT.
Ope)l{.ne scero ycTaHOBHM cJie,nyiOLUYIO npoCTyio JieMMy.
JI e M M a 4. EcJm cpym<QiiH pacrrpe,neJieHHH F (x, fJ) B HCKOTopoif
OKpeCTHOCTH T04KH 00 HMCCT orpaHH'ICHHbiC lJ3CTHbiC npOH~BO).l.Hbie ITO 8,
paBHOMCpHO HerrpepbiBHbi e OTHOCHTCJib!-10 X, If CCJIH OUCHKa (:) rrapaMeTpa 8
TaKOBa, lJTO BCJII1lJHHa
1/N(t9-0o)
rrpH N ~ CXl CJia6o CXO).l.HTCH K HCKOTOpOMY rrpe,neJiy, TO ITOCJIC).l.OB3TCJlbHOCTH
(3) H ( 4) aCHMnTOTI14CCKH 3KBHB3JICHTHbl.
)J,OK333TCJibCTBO 3TOH JICMMbl ITOlJTH 1-ICITOCpC,nCTBCHHO BbiTCKaeT H3 CJIC
,nyiOLUeii.
JI e MMa 5. EcJin ,nJIH rroCJie,nosaTeJibHOCTei'J (6) H (7)
max I g,, - f., I (8)
1 ~.; i< n.~
rrp11 l' ~ CXl CJia6o CXO).l.HTC51 K 0, TO 3TH ITOCJIC).l.OB3TCJibHOCTH 3CHMITTO
THlJCCIUI 3KBHBaJICHTIIbl.
)], o K a 3 a T e JI h c T B o. O oJJO)KH!\I
P,.(a, b)= P {a<.;.,< b, i= l, 2, ... , n.},
P(u, b)= lim P.(a, b),
H (a, b) - TOlJI<a HenpepbiBHOCTH cpyHKU1111 pacrrpe,neJICHH51 p (a, b) .
OycTh
P,(a,b)=P{a :_ ~., < b; i=l,2, ... ,n.}.
Tor.na
P,,(a, b) <: P.(a-E, b+ s) + P {max 1~·, -~ .. , I> E},
• 1< :<,,1'
P.(a+c', b-E') --:-- P ,.(a, b)+ P {max 1~,-.;,, I> c' },
1< i<ll1'
I"}l:C f, e' - I1p0113 BOJibHbiC ITOJIO)KHTCJibllbJC 'IHCJia. J.-13 3THX HepaBCHCTB H H3
ycJIOBIHl (8) CJIC,nyeT
P(a+c', b-e')< lim P.(a, b) < iim P, (a, b) < P(a-c, b+c).
OpmmMaH BO BHHMaHHe uenpepbiBIIOCTb sepoHTHOCTH P (a, b) B pac
CMaTpHBaeMoti TOlJKC, fiO.'l.YlJHi\1 Tpe6ye!\10C.
Bo3npau~aHCh Terrepb K JICMMe 4, ocTaeTCH 3aMCTHTb, liTO c BepoHT
HOCThiO, CKOJ!h -yro,ni!O 6JIH3KOH K 1,
Is, I= I§~- s1 i < NIH-Oo l2 c .
\N
::.. :Y"KpaHHCKHi'l: MaTCM:IT. :m:ypua.'l, T. V, J\~ ·1. 425
B ·CHJIY JI,CMMhi 4 B ,n:aJihHCHIIICM MO)!{HO BMCCTO ITOCJICJJ:OBaTCJihHOCTH (3)
OrpaHH'IHThCH paCCMOTpeHHCM ITOCJICJJ:OB8TCJlhHOCTH ( 4). 3aMCUfM CIII.C, 'ITO
B CHJIY JICMMhi 4 BMCCTO BCJIH'IHHhi VN (li-80 ) B (4) MO)!{HO rrocTaBHTh, He
H3MCHHH rrpe.n:eJibHoro pacrrpe,n:eJieHHH P (a, b), mo6yro .n:pyry10 BeJIH'IHHY,
OTJIH'I810IUYIOCH OT Hee Ha 6eC'KOHC'IHO MaJiyro.
ECJIH npe.n:rroJIO)!{HTh, 'ITO ou.eHKa rrapaMeTpa A rrpoH3BO.D:'HTCH no MeTo,n:y
M8KCHM8JihHOrO npaBJJ:OITO,n:06HH, TO ( CM., Harrp.HMCp, [16])
V}l' (ti- 8 ) = _1_ ~ o log {1 (00 ) _ _ 1_ N' +
• [2 VN' '7 i){) + 7JN'- "VN' fq;' f'J_v·•
r.n:e N' - 'IHCJIO Ha6JIJO,n:enuli, HCITOJih30BaHHhiX ,n:JIH orrp.e,n:eJieHHH H, "'N' -
BCJIH'IHHa, cxo.n:Hr:u.aHC51 no sepoHTHOCTH K 0 rrpH N' ~ =. !; (0) - pe3yJihTaT
no,n:CT8HOBKH i-TOrO H86JIIOJJ:CHHH B •nJIOTHOCTb pacrrpe,n:eJICHH5I f (X, 0) =
iJF(x, 0)
H ox
B COOTBCTCTB'HH C 3THM B ,n:aJihHCHIIICM Mbi ITOJIO)!{HM
(9)
(10)
r ,ir,e rp;- HC38BHCHMhiC MC:tKJJ:Y C06oif, O.D:HH8KOBO pacrrpe,n:CJICHHbiC BCJIH
'IHHbl H
Mrp; = 0, Mrpl = a2 < = ·
06hi'IHO ,n:JIH rrposeprm 3a,n:am-rol1 qJyHKrmH pacrrpe,n:eJie HHH F (x) 11 .n:mr
OU.CHI<H HCH3BCCTHOrO 3H3'1CIUI5I rrapaMCTpa 0 HCITOJih3 YIOTC5I OJJ:HH H TC :tKC
Ha6JI JO)lCHH5l. O,n:J-IaKo He JIHIIICH 3H3'1CIIHH H ,n:pyro}i cJiylJaH, rwr.n:a .D:JIH ou.eH
KH napaMeTpa H rrpoBepKH COrJiaCOB3HI-IOCTH 6 e pyTC5I }J,Ba p5I)l3 HC3 ai3HCl!M b'X
Ha6JIIO)lCtiHH. 0CT8HOBHMC5I rrpC)K)lC BCCrO Ha BTOpoif B031\fO)IUIOCTH, HCCJi e
)lOB3HHC KOTOpoif MQ)!{,CT 6hiTb I1pOH3BC)lCI-IO C ITOMOUI,b!O y:»<C H3BCCTHb!X p c-
3 YJihT8TOB, 6e3 3aTpy)lHCHHH. •
fJOJIOR<HM
N _ , _ oF(x(t), 8,)
o'= N'' t-.F(x, 80 ), 1;-F,(A,), q(t)= 00 · ,
rne x (t)- <PYHKU.HH o6paTHaH t = F (x,l1 0 ). Torna
1, = g(t,) = V N{F,-t,}- (J(wN,; q(t,) ). (11)
f1pH 3TOM Mbl rrpe)liTOJIOR<HM, 'ITO Q (f) HCrrpepbiBHO )lH<fJ<fJepCHU.HpyeMa5!
<fJYHKU.H51 t, H q (0) = q ( 1) = 0.
PacCMOTpHM Terrepb nu<P<fJepeHu.HaJihHOe ypaBHCHHe
au 1 o2u
a~+ 2 iJx' = O;
426
nycrh uT('~, x) - peweHHe :noro ypaBHeHIHI B o6JiaCTH 0 <; ~ < T,
(l+~>[a+o0 (z; q(1 :~))] <x<(l+-r)[b+oo(z; q(1 :~)) l·
o6parn.aiOrn.eecH B 0 Ha KPHBhiX
x = ( 1 + ~) [a + do ( z; . q ( 1 : T} ) ] •
x= (1 +T) [ b+o0 (z; q ( 1 :T))],
H YJI.OBJieTBt>pHIOW.ee yCJioBmo uT(T, x) = I.
B CHJIY npHHII.Hna MaKCHMyMa ,n:JIH ypaBHeHHH TenJIOnpoBO.D:HOCTH noCJie
,n:oBaTeJihHOCTh uT('~, x) npH B03paCTaHHH T MOHOTOHHO y6hiBaeT, H no3TOMY
cymecTByeT
lim uT(T, x) = u0 ('~, x).
T-+oo
N • , J~,-
Teo p eM a 4. ECJIH : I) npH N ~ oo HMeeM N,-cJ. H vN (0-fl0)
CJia6o CXO.D:HTCH K HeKOTOpOMY npe)l.eJiy, 2) <i>YHKII.HH F (x, fJ) HenpepbiBHa
no X H 06Jia,n:aeT HenpepbiBHbiMH H orp aHHqeHHbiMH qaCTHbiM.H npOH3BO)l.
HbiMH no o BToporo n opHJI.Ka, 3) ou.eHKa 0 napaMeTpa 0 u 3MnHpuqecKaH
<i>YHKII.HH pacnpe)l.eJieHH5I noJiyqeHhl no JI.BYM nocJie,n:oBaTeJihHOCT5IM He3aBH
CHMhiX Me:iKJI.Y C060ii: Ha6JIIO.D:eHHH, TO YCJIOBHaH Bep05ITHOCTb COBMeCTHOfO
BblllOJIHeHH51 uepaBeHCTB
a< VN{ F,- F,(ii)} < b, i= 1, 2, ... , n; (12)
npu runoTe3e
VN' (!i-H,) = Z:-,,
r.n:e z_,,__, cxo,n:HTCH no BepoHTHOCTH K z, CTpeMUTCH K u\) (0, O)=u0 (0, Ofz).
,lJ,o Ka3aTeJibCTBO 3TOH TeopeMbl CJie,'LYeT H3 p e3yJibTaTOB, noJiyqeHHbiX
B [11 ]. 0pH 3TOM CXO.li:HI\IOCTb 6y)l.eT paBHOMepHOH OTHOCHTeJihHO Z B JII060M
KOHeqnoM mrrepBaJie -A <; z <; A. l13 r eopeMbi 4 .rrenw noJiyqnTh npe
,n:eJi hHYIO 6e3y CJIOBHYIO BepOHTHOCTb COBI\IeW.eHH51 HepaBeHCTB (12) . 0Ha
pasHa
1 foo ) - 2z;,
11- u0 (0, Ojz e dz.
,2na
(13)
D ep eii:)l.el\r r en eph K 6oJiee cJio:tKHOMY cJiyqmo , Kor.n:a 3MnHpHqecKaH
<i>YHKIJ.HH pacrrpeJI.eJieHHH F; H ou.eHKa napa MeTpa 0 noJiyqenhi no o,n:HHM
H TeM :t~<e p e3yJihTaraM ua6JIIo,n:enuii: . Dpn 3TOM N' = N.
B )l. aJibHeii:weM Mhi o rp amiqHMCH cJiyq aeM OJI.noro napaMerp a . OcnoBHoii
p e3yJihT3T, KOTOpbiH 6y)l.eT HmKe ycraHOBJieH, MO)IUIO c<t>opMyJIHpOBaTb CJie
JI.YIOJL{HM o6pa30M.
T e o p e M a 5. Dpe,n:noJio:iKHM, qro :
a) OII.eHKa HeH3BeCTHOfO napaMeTpa npOH3BO.D:HTC5I TaKHM o6pa30M, qTO
V-- 1 N
N(0-00 ) = V N;!i'<p(§,) + fJ(N),
427
I',LI,C { ~;} - 'nOCJIC,LI,OBaTeJihHOCTb B3aHMHO HC33BHCHMbiX H3MCp,CiiHi'l CJiyqal'f
HOH BeJIHLfHIIhi ~. CJiyLiaiiHaH BCJIHLfHIIa rp U) HMeeT KOHClJHbiii MOMCHT BTO
poro rropH,LI,Ka, rrpHLieM
Mrp(~)= 0, Mrp2a)= a2,
H 7] (N) CXOJ~HTCH ITO BepOHTHOCTH K 0;
b) K CJiyqaiiHOH BCJIH'IHIIC 1/J(~) = 1p(x(~)), I',LI,C x(f)- cpyHKUHH, o6paT
HaH cpyHKUHH t = F (x, 00 ), rrpHMCHHMa rrpe,LI,eJibHaH TeopeMa JI..TIH IIJIOTHO
CTel'f;
c) cpyHKUHH F (x, f!) HerrpepbiBIHl no x 11 o6Jia,LI,aeT HenpepbiBHhiMH orpa
HHLfeHHhiMH IIpOii3BO,LI,Hb!MH IICpBOI'O li BTOporO IIOpH,LI,Ka.
00JIO)KHM
a (t) = J tfJ (~) d~, q (t) = a~ F(x (f), 0) lo=o ,
I 0
0
A(t, s, afz) = [(1-t) (z-a)-(s+zq(~))ct. (t)][(1-t)a' (t)+rt (t)] __
(1-t) [(1-t) J a'2 (t)dt-cx 2 (t)]
t
s+ zq(t) , ()
- 1-t - zq l,
U 0 (t, s, a)= lim ua(f, s, a),
a--+1
r.rr,e ua(t, s, a)- pewenHe rrapa6oJIHLJecKoro ypanHeHHH
B 06JICJCTH 0 <; t < a < 1, - ex:> < a < + ex:>, Gt ·< S < a2, a1 < 0 < G2,
YJI.OBJICTBOpHIOLUee ycJJOBHmr u(a, s, o) == I, u(t, a1, o) = u(t, az, a) = 0.
Tor,LI,a yc.TioBHaH nepoHTHOCTb npH mnon' 3 e VN(U-00)=zN conMeiii,eHHH
HCpaneHCTB
i=1,2, ... ,n,
CTpCMliTCH K Uo (0, 0), K0f')J:3 max fJ;-:> 0 H ZN CXO;'~HTCH 110 BCpOHTHOCTH K Z.
J ... :;. ; n
nycTb ~ - C.TJyqaihiaH ne.T!HlJIUia c <pyHKUHeii pacnpCJ! C,TJ('HHH F (x, flo)·
TorJia rt == F ($;, 00 ) - c.nyqaiiHaH BeJml!Ima, paBIIO~repno pacnpe.rr,eJICHHaH
na m!Tepna.ne (0, I). BBe JICM CJiytiailuyio <j.Jymm,HIO
( ) _ J 1, Ti < f,
x -r{, t -to,
'[t > t.
Tor.rr,a B ( 11) Mhi MO)KCM noJIO)KHTh
- _1 ~ ( t) Fi - N ,..;.. X Tr, t ,
r o7}
( 14)
TaKHM o6pa3oM, ~ = l
-; (t,)= Y N{ _!_ i X(~., t,)- f,}- qj Nq(t;).
N r=t
11MeeM, ,n:aJiee,
a(t)=Mx(-r"' t)tJ!(~r) = J?/•(-r)d~, ct(l)=O,
0 -
{J(t)=Dx(-r" t) 1/J(Tr) = J 1/JZ(r)d~·-a'(t).
0
Bse.n:eM eme nocJie.n:osaTeJihHOCTb BeJJH'IHJI
~ (t1)= V N {l _!_ i X ( Tr, / 1) 1/J ('t:r) - IX (f,) 1l.
Nr=l
OtieBH.D:HO,
7J (1) = a>.v·
J1 eM M a 6. DpH mnoTe3e
1](1)= a>N=z
nOCJie,ll.OBaTeJJbHOCTb JI.ByMepHbiX BeKTOpOB
act,),~(t,); i=1,2, ... ,n}
CBH3aHa B npocTyiO u.enh MapKoBa.
(15)
(16)
)leH:cTBHTeJJhHO, ecJJH 7J ( l) = z 3a,n:aHo, To, 3HaH Ht;) = s, JierKo HaH:nr
lJHCJIO CJJYli3HHbiX BeJJHlJ"Hl-1 T,, 'HMCIOJJ.J.HX 3 H3'!eHH5I, He npeBOCXO,ll.HW.He f;. 06o-
3HalJ35I ero qepe3 v8 ( t;), noJJytiHM
~·.(t,)= YN[s+ zq(t,)]+ Nt,.
K:poMe TOrO, ecJIH H3BeCTHO
TO BC5IK35I ,ll.OnOJJHHTeJJbH35I HH4JopM3U.HH, KOTOpaH MO)!{eT 6bi:rb noJJy'!eHa
J-13 3H3HH5I BeJ!HlJHH .;;(tj), 'E:(f1), fi<f1 , CBO,ll.HTCH K HllcpOpMaU.HH 0 pacnpe,ll.e
JieHI:l.H p .e3yJJbT3TOB H36JJIO,ll.eHHH, nonaBUIHX B nepBbJe i HHTepBaJJOB rpynm.f
pOBKH, H n03TOMY H e OK33bJBaeT ,ll.OnOJJHHTeJJbHOrO BJJH5IHH5I Ha pacnpe,n:e
JieHHe BCJJH!JHH '[(tj), rj(tj ) npH j> f.
)lOK333HII35I JJeMMa OTKpbiBaeT B03M0)!{1·10CTb H3ytieHH5I npe,n:eJibHOrO
YCJJOBHOrO pacnpe,n:eJICHHH 3KCTpeMaJJbHbiX 'IJJ,eHOB nOCJJe,ll.OBaTeJibHOCTH
(~ (/1)} C nOMOJJ.J.biO MeTO,ll.a ,ll.HcpcpepeHU.H3JlbHbiX ypaBHeHHH [I 0, 11). )lJIH
npHMCHeHHH y)!{e H3BeCTHbiX pe3yJJbT3TOB HaM nona,n:o65ITC5I Ol.l;e HKH ,ll.JIH
yCJioBHhix MaTeMaTHtieCKHX O)!<H,ll.aJmH: npnpam:enHH:
npH 33,ll.3HHbiX
Lff_(t1) = ~(tt+ 1)- §(11),
LJ;j(t,) = fj(t,+l)- 'ij(t,)
429
PaccMOTpHM Terreph CJIC.D.YJOlli.YJO BcrroMoraTeJibHYJO 3a)l.a'ly. Oycrh
{ Tk} - IIOCJIC)I.OB3TCJibHOCTb HC33BHCHMbiX paBHOMepHO pacrrpe)l.eJICHHbJX Ha
HHTepBaJie ( o, T) CJiy'laHHhiX BeJIH'IHH. OoJio:tKHM
r)l.e
1 N
T/2 (t) = 11 - - _l7 {X (-c-"' t) t/J (•.)- a (I)},
yN r-1
t
a (t) = Mx (T., t) 1/J ( r.} = ~ J 1/J ( T) dx,
()
t
{J (l) = M']~ (t) = ~ J tf-'2( -c-) dr- a! (i).
0
Tpe6yeTCH oiJ,eHHTh yCJioBHbie MOMeHThi rrepBbiX .D.BYX rropHJl.KOB BeJIH'IHH
7JI (t), 172 (t) .D.JIH MaJibiX t rrpu mrrOTe3e "12 (T) = z. DpH 9TOM Mhi 6y.n.eM
npe)I.IIOJiaraTb, 'ITO l/J (Tr) - a6COJIIOTHO HerrpepbiBHa (T. e. HMCCT IIJIOTHOCTb
pacnpe)l.eJiemm) H y.n.oBJieTBOpHeT Tpe6oBaHHHM np,e.D.eJihHoli TeopeMbi .D.JIH
IIJIOTHOCTCH CYMM HC33BHCHMb!X O)I.HHaKOBO pacnpe)l.CJICHHb!X CJiy'laHHb!X BC
JIH'IHH. Tor.n.a K "12 (T) 9Ta npe.n.eJihHaH TeopeMa rrpHMeHHMa, H IIJIOTHOCTh
pacrrpe)I.CJICHHH p2 (T, z) BCJIH'IHHbl 1]2 (T) paBHOMepHO OTHOCHTCJibHO Z CTp e
MHTCH K rrpe.D.eJIY
1 -~
~===~e 2fi< Tl,
V 2n{J(T)
BBe.n.eM eme yCJioBHYIO IIJIOTHOCTh P2 (T, zIt, s, a) BCJlH'IHHhi "12 (T) rrpH
nmore3e "'I {t) = s, "12 ( t) = o. Tor)l.a, ecJIH f (x, y)- rrpoH3BOJihHaH nerrpe
pb!BHaH <}lyHKli,HH rrepeMCHHb!X X H y H CCJIH M {f[TJI (t), "12 (t) ]/T, z}- ycJIOB
HOe MaTCM3TH'ICCKOe O:ti<U)l.3HHe BCJIH'IHHbl /['f}I (f), 'f} 2 (t)) rrpH rHIIOTC3e
1]2 (T) = z, TO H3 <}lopMyJihi 6aiiecca cJie.n.yeT
M{f[rlt(i), 'rl2(t)]/T, z} = Mf["J1 U), "'2(t)]r['lldi), 1]2 (t)], (17)
( ) _ p 2 (T, z/t, s, a)
r s, a - (T ) . P2 , z
3ai'!MeMC5I rrpe)!{)I.C BCero BCJIH'IHHOIO ')' ( S, a).
BeJIH'IHHY 172(T) rrpH mrrore3e TJ 1(t)= s, TJ 2 (t) = a MO)!{HO OTO)!{,ll;CCTBHTh
C 9KBHB3JICHTHOH BCJI.H'IHHOH
r)l.e
1 N-•·. T-t -
r, 2 (T)= V N ·~ 1/J(r.')- -;y- VN a(t, T) +a,
T
a(t, T) = T~ t f t/J(T) dr,
t
, v- t S=-'!...- N-
VN T
H seJIHIJHHbi 1:/ pasHoMepHo pacnpe.n.eJie Hbi R3 HHTepB3Jie {t, T) . floJioMHM
1 N-•s
1)2, = v- .£ ['l'<"'r'>- Ml/1(7:/)); os>
N-l'• ,
Tor.n.3
T
Tlt' = v N~: [ '12 (T)-a + T~ f f 'I' (7:) d7:],
t
T
D"J/={J(f, T) = T~f f tp2(-,;) d~-a2(f, T).
B CHJIY npe.n.eJibHOI'i: TeopeMhi ,UJIH IJJIOTHOCTel'i: nJIOTHOCTb p3cnpe.n..eJieHH5I se
JIHIJHHbi r;2' p3BHOMepHo OTHOCHTCJibHO t, 0 < t < T' < T CTpeMHTCH K
1 ·- --~-v 2n{J <CT) e 2,8 (t, TJ,
OTKY.ll.3
1
( T-f)-- 2 f (z-a+scx (f, T) )2 T}
p,(T, z/f, s, a)- 2n{J(t, T) T expl- 2{J(f, T) (T-T ·
p3BHOMepHO OTHOCHTCJ!bl-10 S, o, - 00 < S , o < + 00 H f B p3Hee YK333HHOM
Hineps3Jie. 0TCIO.U3 y)Ke cJie.n.yeT
[ /'I (O,T)T ] } { (z-a+scx(t,T)) 2 T z 2 }
l ·(s, a)- /] (t, T) (T- t) exp - -cxjJ (f, 'F) (T-f) + 2{J(O, T) = ro(l)
p3BHOMepHo OTHOCHTeJibHO t, s, o H z, rne I z I <A, A - JIK:>6oe > 0. B IJ3CT
HOCTH, BCJIHIJHI-13 y ( S, a) p3BHOMepHO orp3HHl!CH3 B YK333HHbiX npe,UCJI3X.
flpH M3JlbiX f
y,(t} = 1 + z(<T-Sct(O, T)) + 0(t)+(a'+s2)0
fJ~n ,
r,n.e 8 - orpaHHIJCHI-135! <f>YHKU:H5I.
flycrb X, =X, ("J1, ' i2) - <f>YHKU:HH, p3BH35I 1, CCJIH I '71 I< e H I '1]2 I < e,
H p3BH3H 0 B rrpoTHBHOM CJIYl!3e. Tor.n.3 ;lJIH . ,ype33HHbiX" MoMeHTOB BCJIHIJHH
']t(f) H 7J2{f) 6e3 Tpy,n.3 IJOJIYIJHM
z
M {X,1'/z (f)/ T, z} = {J(O, T) M('l; (t)-a (0, T)"J1 (t) TJ2(f)) + o2 (e, f),
M{X,11 ~ (t) / T, z} = JJ;f"l~ (t) + o3 (e, f),
M {X, "'t (f)'l'}, (f)/T, z}= M'/1 (t)7J2 (f)+ cf.l (e, t),
M{x,TJ ; (t)JT, z}= llb1! (t) + o5 (!, f),
r,n.e BCJIHIJHHbl ~i(e, f) T3KOBbi, IJTO
I. -~.- o,(e, f)
1m 1m-- =0. ·--o t-+O t ·
(19)
431
3aMeTHM eme, 'ITO H3 paBHOMe pHOH orpaHH'lei-IHOCTH BeJIH'lHHbl Yo (f) CJie,ayeT
P{max[J7h(i)J, I7J2 (t) JJ> c/T,z}
< CP {max [I 1}1 (t) J, I 11~ (t) I] ::> t} = o, (t), (20)
r,n;e o, (f) / f -c> 0, KOr,aa t-0 rrpH JII060M <jJHKCHpOBaJIHOM e > 0. Jlmi 6e3-
YCJIOBHblX MOMeHTOB BeJI"H'lHH 'Y};(f) HMeeM CJie,n;yiOmHe BbipmKeHH51:
t t t
M"tj; (t) = T, M1Jl (t) '72 (f) = T l/1 (0)+ 0 (t), M7]: (t) = T 1/!2 (0) + 0 (f). (22)
Bo3spaTHMCH K BeJIH'lHHal\1
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1
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f;
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- _ 1 jN J _ _ s + zq ( t,) (t ) }
z- V N. l z a I - t, a ' . (23)
BeJIH'lHHa LU (t;) MO)I{eT 6biTb rrpe,n;cTaBJieHa B BH,ae
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N. • '
s+zq(t1)
- 1_ 11 Lit,- zLlq(t,). (24)
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1
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I
AHaJIOI'HlJIIO IlOJJYlJaiOTC51 OCTClJibHbiC ,ype:3ClHHbie" MOMCHTbl n epBOfO H BTO
poro nopHJJ.Ka BeJJHlJHH LIH f;), Ll'7 (t;) _ B CHJIY yH<e UHTHponmiHbiX p e 3yJ1 b
TaTOB pa60Tbl [11] H 11pl!HHM351 BO BHHMCIHHC 33MClJ3HH51, C)J.CJI3HHL>IC Heno
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BOCTH TeOpCMbl 5.
JI11TEPATYPA
I. B. B. fHeJJ.eHKO n B. C. MHxaJieBH'I, ,UAH CCCP, 82, .N!! (), 1952.
2. K. L. C h u n g- and W. Fe II e r, Proc. Nat. Ac. Sc. 35 (1949).
3. M. Lip chit z, Proc. Am. Math. Soc., v . 3, N~ 4 (1952).
4>. B. B. ruc.n.eHKO H B. C. MHXa Jie llH'I, ,UAH CCCP, 85, N2 I, 1952.
5. B. B. r He .n. e H K o H E. JI . P sa 'I e sa, ,UAH CCCP, 82, N~ 4 (1952).
G. H. B. C M H p u o ll, firoJIJI. MocK. roc. yu-Ta, B. 2 ( 1939).
7. A. H. K o JIM oro poll, Giornale Instil. Ita!. Attuari, 4 (1933).
8. W. F c II e r, Ann. Math. Statistics, 22. N~ 3 (1951).
9. D. B. r II ell. e H K 0 H A. H. K 0 Jl M 0 r 0 p 0 ll, npe.n.eJibHhle pacnpe.n.eJieHH!I
,LJ.JI!I cyMM He3allHCHMhiX CJiytJallllhiX llCJIH'IHH.
10. A. 51. Xu H 'I u H, AcHMnTOTH'IeCKHe JaKOHhl TeopHH nepoHTHOCTeii, OHT11.
1936.
II. 11. 11. r H X M a II, MaT. c6. KueiJCKOro yu-Ta, N~ 8.
12. 11. 11. r H X M a H, ,UAH CCCP, 82, N" 6, 19512.
13. J . L Doob, Ann. Math. Statistics, 20,3 (1949).
141. D. B. r He .D. e H K 0, Kypc TeOpHH nepo!!THOCTCH, f11TT JI, 1950.
15. Z. W. Birnbaum, Journal Am. St. Assoc. , v. 4'7, N~ 2-59(1952).
16. r. K paM e p, MaTeMaTJi'ICCKHe MeTO.D.hl cTaTHCTHKH, f1111JI, 1948.
noJiy'leHa 17 HIOH!I 1953 r.
Knen.
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| id | umjimathkievua-article-7755 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:40Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e6/5ac8c731ebcc7aa956a6134efad952e6.pdf |
| spelling | umjimathkievua-article-77552023-08-15T10:11:31Z О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики Гихман, И. И. Гихман, И. И. Настоящая статья состоит из двух частей. Хотя задачи, рассматриваемые в них, на первый взгляд, кажутся мало связанными, в действительности же их объединяет общность метода исследования и внутреннее единство. В первой части доказываются некоторые теоремы, относящиеся к условному распределению функционалов от последовательности независимых случайных величин, во второй — рассматриваются вопросы, связанные с распределением колмогоровского критерия согласия в том случае, когда проверяемая функция распределения содержит параметры, определяемые эмпирическим путем. Настоящая статья состоит из двух частей. Хотя задачи, рассматриваемые в них, на первый взгляд, кажутся мало связанными, в действительности же их объединяет общность метода исследования и внутреннее единство. В первой части доказываются некоторые теоремы, относящиеся к условному распределению функционалов от последовательности независимых случайных величин, во второй — рассматриваются вопросы, связанные с распределением колмогоровского критерия согласия в том случае, когда проверяемая функция распределения содержит параметры, определяемые эмпирическим путем. Institute of Mathematics, NAS of Ukraine 1953-11-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7755 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 4 (1953); 413-433 Український математичний журнал; Том 5 № 4 (1953); 413-433 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7755/9443 Copyright (c) 1953 И. И. Гихман |
| spellingShingle | Гихман, И. И. Гихман, И. И. О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_alt | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_full | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_fullStr | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_full_unstemmed | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_short | О некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| title_sort | о некоторых предельных теоремах для условных распределений и о связанных с ними задачах математической статистики |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7755 |
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