О роли максимального слагаемого при суммировании независимых случайных величин

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Збережено в:
Бібліографічні деталі
Дата:1953
Автор: Гнеденко, Б. В.
Формат: Стаття
Мова:Російська
Опубліковано: Institute of Mathematics, NAS of Ukraine 1953
Онлайн доступ:https://umj.imath.kiev.ua/index.php/umj/article/view/7734
Теги: Додати тег
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Гнеденко, Б. В.
Гнеденко, Б. В.
author_facet Гнеденко, Б. В.
Гнеденко, Б. В.
author_sort Гнеденко, Б. В.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2023-08-15T10:12:02Z
description -
first_indexed 2026-03-24T03:33:14Z
format Article
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B pa6oTax [2] H [3] .LJ:OKa3aHo, qTo npe.neJibHhiMH 33KOH3MH .LJ:JIH <PYHK­ UHH pacnpe.neJieHHH seJIHqHH 1].- A B.,. n npH H3).1.JieMameM no.n6ope noCTOHHHhiX B,, > 0 H A 11 MoryT 6h!Th TOJihKO pacnpe.neJieHHH r.ne a- TIOJIOMHTeJibH3H TIOCTOHHH35I 'H npH x<:O, npH x>O; npH x <;:: O, npH x> 0; A (x) = e-·-x. B HaCTOHLUeA pa6oTe Hac 6y.nyT HHTepecosaTb TOJihKO pacnpe.neJieHHH !lia(x) H A (x). Teo p eM a B. LI.JIH Toro l.JT06bi npH Ha.nJie)J<ameM no.n6ope nocTOHH· HhiX B,11 (nocTOHHHbie A 11 MOMHO Bbi6paTh paBHbiMH 0) <PYHKU.HH pacnpe.ne- JieHHH BeJIHlJHH 7Jn -A CXO.LJ:HJIHCb K pacnpe.neJieHHIO l])a (x ), He06XO).I.HMO B,. " H ).1.0CT3TOlJHO, l.JT06bl npH Jl!060M k > 0 BbiTIOJIHHJIOCb COOTHOUieHHe . 1- F(x) ~1__?!, 1- F(k-~) = k a. (8) nocTOHHHbie B,. onpe.neJI HIOTCH KaK H3HMeHbUIHe KOpHH HepaBeHCTB 1 1-F(x(l +0)) <-< 1- F(x(l-0)). . n {9) T e 0 p eM a r_ EcJIH <PYHKU.H5! pacnpe.neJieHH5! 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B CHJIY TeopeMbi B, KaKOBhi 6bi HH 6biJJH TJ > 0 H fl > I, MO)l{HO HaihH TaKOe Xo, 'ITO 1-F(k' x0 ) (l+ )·ka 1 - F(k'+l xo) = e, ' npH'IeM I c5 I ,.;;; TJ npH Bcex s (s = 0, I . 2, .. -) . Mbi npe.n:noJJO)l{HM, 'ITO (I - TJ) k• > I. QqeBH.LI:HO, 'ITo 00 J I X Ia-• dF(x) = ~ J xa- • dF(x) <. x 0 lr.3-Ix0 .< x < kBx0 <. x~-· [1-F(x. )] i: k' ta-•) 1- ~(k'-1 Xo) <. k"x~-· I(' 1 .)s-1. s=l 1 F(xo) s=l (1- 1J) k B CHJJY Bbi6opa TJ H k p5I.n: B npaBoi1 qacTH HepaBeHCTBa cxo.n:HTC5I; nepBaH 'laCTb TeopeMbl TaKHM o6pa30M JJ:OK83 8H8. TaK KaK Joo \X \"+• dF(x) ~ ~ k(s-l) ta+•Jxa+•[l-F(k•+1x )] [ 1- F(k' xo) -1] :::;;:, ·7 ..::.. 0 0 1 - F(k•+l X ) 7 •=1 0 n 1 ~ 1J > I - 1], TO 1 !' 1) > (1- '7) k ' > 1 n, CJie.n:oBaTeJibHO, p5IJJ: B npaBoa 'laCTH HepaBeHCTBa p acxo.n:HTCH. T eop e Ma .n:oKa3aHa. 294 .[{ 0 K a 3 a T e Jl b C T B 0 T e 0 p e M bl 2. 0lieBH,li,HO, liTO CJie,LJ,yeT pac­ CMOTpeTb TOJibKO cJiyliaH, KOr,LJ,a F(x) < I npn Bcex x. Mbi Haxo,LJ,HMCH B YCJIOBHHX TeopeMbl r. 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B CHJIY (9) HOpMnpyiomne K03cpcpnu,neHTbi Bn ,LJ,JIH cxo,LJ,HMOCTH pacnpe- " -~n ,LJ,eJieHHH BeJIHliHH K npe,LJ,eJibHOMY ,li,OJI)!{Hbl 6biTb Bbi6paHbl K3K HaH- MeHblllHe KOpHH HepaBeHCTB 1-F(x(l +0)) + F(-x) -.< !_ _,;;;; 1- F(x) + F(- x(l +0)). n CorJiacHo (6) TOliHO TaKHM *e nyTeM onpe,LJ,eJimoTCH H HopMnpyioiu,ne K03cfJ­ cpHu,HeHTbi Bn ,LJ,JIH cxo,LJ,HMOCTH cpyHKU.HH pacnpe,LJ,eJieHHH BeJIHliHH (2). I ) B nocJie.nHeM cJiy<Jae npe.nnoJiaraeTcH, 'ITO n (3) c =F oo . 295 ECJIH C =/= oo, TO B CHJJY (3) COOTHOlllCHHC (4) npHBO,[(HTCH K BH,ny (8), a COOTHOli,ICHHC (6) K BH,ny O+c) (1-F(x(l +0))1 < _!_ < (1 +c)[l-P(x)] n HJIH, qTo TO )!(C C3MOC (aCHMITTOTHqeCKH), K BH,ny _!_ 1 _!_ 1-F((l +ct x(l +O))< - < 1-F((l +c)" x). n . Mbi BH,nHM, TaKHM o6pa30M, qTo HOpMHpyiOmHe Ko3<tJcpHU.HCHTbi Bn ,nm1 CXO,[(HMOCTH <tJyHI<I.J.HJ{ pacnpe,neJJCHHH CYMM (:2} JIHlllb ITOCTOHHHbiM MHO)!(H­ TCJICM OTJIHqaiOTCH OT HOpMHpyiOIJ..I:HX K03<tJ<tJH!J.HCHTOB, C KOTOpbiMH <tJym>- ?j,. 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CMHpHOBa, co,nep)KaW,HXCfl so BTO­ poii qaCTH pa60Tbl [6], MQ)KHO Bb!BeCTH p5I,n Teop eM, no,no6HbiX TeM, KOTOpbie ,!I.OI<a3aHbi B HaCTOHW,eM naparpacpe. Ll.JIH HJJJIIOCTpall,HH Mbl DpHBe,D.eM cpOp­ MYJIHpOBKY o,nHoro TaKoro npe,nJIOR<eHH51. Pa c noJIO.I!<HM BeJI Hqm·Ibi I ~1 I, I ;~ I, ... , I t,.l B nop5I,nKe Bo3paCTaHH51. 06o3HaqHM h-y10 BeJIHqHHY cnpasa qepe3 ?;,~kl ; 5ICHO, qTQ (~1 ) = Sn· Teo p eM a 5. EcJIH cpvHKll,Hfl pacnpe,neJi e HH51 F (x) npHHa)l)leR<HT 06JI3CTH npHT5I )KeHH51 YCTOi1qHBOrO 33KOH3 C xapaKTepHCTHqeCKHM DOK333Te­ JieM a =/= :2, TO C TeMH )Ke C3MbiMH HOpMHpyiOW,HMH MHOR<HTeJ151~1H B 11 , KaK H ,li.JI51 CYMI\1 (2), CXO,li.5ITC5l K C06CTBeHI-IOM)' npe)l.eJi hHOMY pacnpe,neJieHHIO ( (k) ¢YHK11.HH pacnpe,neJJe iiHfl BeJinquH ., ; npc,neJibHoe pacnpe,neJierme onpe,ne­ B,. JI5IeTC51 paBCHCTB3MH (J)~k) (x) = 0 ,D.JI5l x < 0, xa (J)(k) (x) = 1 J zk-1 e-z dz ,ll.JI5I x > 0. a (k-1)! 0 JU1TEPATYPA L G. B. r He,!{ e H K 0, I( TeOpHH oOJiaCTCfl rrpHT5DK.eHHH YCTOll'llillblX 3aKOHOB, Y'le­ Hhle 3311HCKH MOCKOBCKOrO ymmepCHTCTa, Bblll. 30, 1939, 61-82. 2. B. B. r Hen e H K o, D pc;LC,1bllb!e TeopeMbl J\JIH MaKCIIMaJibi!OfO '!JICHa BapnauHOH­ Horo pHna, ,ll.AH CCCP, 32, 1941. 3. B. B. r 11 e !L e H I< o, Sur la distribution limite du terme maximum d'une serie alcatoire, Annals of mathematics, 44, N~ 3, 1941, 423--453. 4. B. B. r H c ,!{ e H K 0 I! A . H. I( 0 Jl M 0 r 0 p 0 ll, ITpe.neJibHble pacrrpe.neJieHIIH ,l{JIH C)'MM IIC3aJJIICHMb!X CJry•IaiiHhiX DCJIH'!HH, fTTJ1, 1949. 5. W. Doe b 1 in, Sur !'ensemble de puissances d'une loi de probabilite, Studia Math., T. IX, 1!940, 71-96; Ann. Ecole Norm., 3, v. 63, 1947, 317-350. 6. H. B . C M H pH o D, ITpe:J.e,1bHuie 33KOIIbi pacnpc,a,eJieiiiHI ,l{J1H 'l.neHOD napnaunon­ uo ro pH,tLa, TpyJJ,bl M aTcMaT H'lCCKoro nu-Ta HM. B. A. CTeKJ10ila, XXV, 1949. 7. A. 5I. X H n '!II 11 , Dpc.'l.CJiblluie 3aKOIIul /LJ15! cyMM IIe3anncnM~>rx cJiy'lai'w~>Ix neJIH­ 'lHH, fOHTH, 1938. n o.>ry•ICHa 24 QJellp ami 1953 r. I(HCD 0291 0292 0293 0294 0295 0296 0297 0298
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institution Ukrains’kyi Matematychnyi Zhurnal
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spelling umjimathkievua-article-77342023-08-15T10:12:02Z О роли максимального слагаемого при суммировании независимых случайных величин О роли максимального слагаемого при суммировании независимых случайных величин Гнеденко, Б. В. Гнеденко, Б. В. - Institute of Mathematics, NAS of Ukraine 1953-08-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7734 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 3 (1953); 291-298 Український математичний журнал; Том 5 № 3 (1953); 291-298 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7734/9431 Copyright (c) 1953 Б. В. Гнеденко
spellingShingle Гнеденко, Б. В.
Гнеденко, Б. В.
О роли максимального слагаемого при суммировании независимых случайных величин
title О роли максимального слагаемого при суммировании независимых случайных величин
title_alt О роли максимального слагаемого при суммировании независимых случайных величин
title_full О роли максимального слагаемого при суммировании независимых случайных величин
title_fullStr О роли максимального слагаемого при суммировании независимых случайных величин
title_full_unstemmed О роли максимального слагаемого при суммировании независимых случайных величин
title_short О роли максимального слагаемого при суммировании независимых случайных величин
title_sort о роли максимального слагаемого при суммировании независимых случайных величин
url https://umj.imath.kiev.ua/index.php/umj/article/view/7734
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