Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations

By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.

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Дата:1999
Автори: Dorogovtsev, A. A., Дороговцев, А. А.
Формат: Стаття
Мова:Російська
Англійська
Опубліковано: Institute of Mathematics, NAS of Ukraine 1999
Онлайн доступ:https://umj.imath.kiev.ua/index.php/umj/article/view/4590
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Dorogovtsev, A. A.
Дороговцев, А. А.
Дороговцев, А. А.
author_facet Dorogovtsev, A. A.
Дороговцев, А. А.
Дороговцев, А. А.
author_sort Dorogovtsev, A. A.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2020-03-18T21:09:14Z
description By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.
first_indexed 2026-03-24T03:01:46Z
format Article
fulltext Y~K 519.21 A. A. ~oporonueB (HH-T MaleMaTHKH HAH YKpaHHbl, KHeB) MEPhl IIOCEII[)EH.H~ H 3PFO~HqECKAH T EOPE1VIA ~YI~I IIOCJI~o~[OBATEJI]bHOCTH HTEPAIIHH CO CJ1TqAHHbIMH BO3MYIIIEHHHMH By using local visitation measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem. 3a/lonoMoro~ JIOKa.IIhHHX Mip nepe6ynaHH~l onrlcaHo rpatlHqHy noBr noc.ni/IoBnoeTi iTepatfifl a BHIIa/IKOBHMH He O/IJtaKOBO pO3rlO/lidleHHMH 3~ypeHH~IMH. ~]K HaCdli/~oK OTpHMaHO BapiaHT dlOKadlbHOi epro~HqHOi TeopeMH. 1. M e p ~ n o c e m e u u a . B/IaHHOI~ pa6oTe Hcc~e~yeTca noBe~eHHe peKyppeHTH~X noc.ne~oBaTe.rmHocTei~ Brtaa x,,+l = ~ ( x , , ) + ~ n , n>_.0, x0 e IR a. (1) 3/~ecs tO : ~ d ---> ~ d HenpephtBHa..,t di)yrlKI~Ha, {~n; n > 0} ~ noc.rle]IoBaTeJ'lb- HOCTb HeaaBHCHMb~X cJIytlaflHl~X BeKTOpOB B ]R d 3a~aHHHX Ha BepOJ~THOC'I'HOM npocTpaacTBe (f~, F, P ) . HJIa noc~e~oBaTeJmHoc'rH (1) ~HO~XeCTBO qaCTaqHUX npe~e.noa n qaCTOTa nocemeHna HX oKpecTHOCTelt MoryT 6u'r'b OnHcaHu a TepMmiax Mep rIocemeaHa, BaeaeHmaX B [ 1 ]. HOaTOMy aTOT nyHKr co~ep~HT onpezle.nenHa H ~a~ ' I '~ Ha [1], Heo6xo/II4M~e B z~a.rmHelttue~. HycT~ { u n; n > 0 } - - ~eTeprdHHHpoaaaHaa noc.ne~tOBaTe.ru, Hoc'rt, a,neMeHToa Ha 1~ a Onpe~ae~enHe 1. C~tumatou4ea ~epo~ nocneOo~arneabnocrnu { Un; n >_ 0} 0n~ n > 0 naz~oaemc~ ~epa n V n = ~_~ ~u t , k=O zcge 5 u ~ ~tepa ecgunu,,mo~ ~taccbt, cocpecgomo~enna.a o motw.e u. Onpe~ae.aen,e 2. l loc.aeSo~zmenbnocmbrone,mb~x~tep { l . tn ;n > 0 } t-ta ff- anee@e 5ope,ae~rux ~mo.~ecmo npocmpancmoa ~ d na..v~oaemca oospacmwou~e~, ec,au 1) 0.a~ npouaoonbnoeo 5openeocroeo A ~ ~ d nocaeSooamenbnocm~, { I.tn ( A ) ; n > 0 } ne yS~,toaem, 2) lira t t n (~ d) = +~'. 11 -"@ r Onpe~te.neHHe 3. T o , ~ u e ~ a naz~aemc .~ mo,oco~ pocma oospacmatou4e~ nocneSooament, nocmu ~tep {P'n; n > 0} , ec.au V e > O: lim gn(B(u, e)) = +**. rl '-') r ,Y~geCb B(u , e ) ~ o m r , pb~n~tl map pazguyca e c ~r o u. MHO3KeCT~O BCeX TOqeK poeTa sospacTa~ome~t nocae~onaTe.rmHoc'm Mep { P'n; n --> 0 } 0603Ha~HM ~ . ~1"o 3a l~ayToe MS, toY~eeTBO. EC.nH { V n ; n ~ 0 } ~ noc- �9 A. A. ,~OPOFOBH, EB, 1999 1SSN 0041.6053. Y~p. ~tam. ~.'ffm,, 1999, m. 51, N e I 123 1 2 4 A . A . ~[OPOFOBI/EB ae~ol;aTeJn, HOCTb C~IHTalOIImX Mep IIOCne~oBaTen~Hoc'rH { u n; n > 0 }, TO ~ ff - - ~IO~KeCTBO q a ~ u x npeAenoB { Un; n > 0 }. Onpe~e~eHHe 4. ]lee noc.~eOoaame.~bnocmu oozpacmamu4ux ~ep { ~n ; n > 0 } u { Z . ; n > 0 } ~ouaanenmn~, ecau 2) c~.4~.~w6b~ o?.panuqewa, ur om~pt, tmb~ ~ o w ~ r U I u U 2 ma~u.x, qmo 3a- ~,~o~a.~e ~ ~ U2, Um ~ 0 , l im ~'n(U2) > 1, lim Zn(U2) >_ I. n --.>"'~ ~,,n(Ul) n .--I."'-~ ~l.n(Ul) Onpe,~e,.aeHHe 5. llocnecgoeamenbnocmb ~,ep, ~eueamenmna~ nocAecgoeamenb- nocmu c,~uma~ou~ux ~ep nocaeOoeame.~bnocmu { u n; n > 0 }, na .~aaemc~ nocneOo- ~ameabnocmwo ~ep noceu(enu,~ { u n; n > 0 }. B [1]/xoxa3ama caow~oumo yTBCp~0mla. TeopeMa 1. Ecau aozpacma~ou4a~ noc.~eDoaamenbnocmb :~ep { Dn; n > 0 } ~r- auaa~enmna nocneDoaamenbnocmu { CnZ ; n > 0}, ebe { Ca; n > 0} ~ nucnooa.~ nocneOoaamenbnocmb, a ~. ~ a e p o m n n o c m n a . ~ ~epa na ~a , mo ~epb~ { ~ln/ Cn; n > 0 } cxoS,~'w.~ caaSo r ~,. TeopeMa 2. Hycmb { ~l n, n > 0 } ~ ozpanuqenna~ noc,~eDoaamenbnocmb ne~a- a u c u ~ x c,~yna~ntax oermopoo o ~e . TozSa cyu4ecmoyem :~mo~cecmoo f2 o c f2 maroe, ,ano P ( ~ o ) = 1 u ~n,~ .~m6oeo co r ~ o nocneSooamenbnocmb { i =~o P~I!-D; n > 0 } (2) . ~ o ~ m c ~ noc.~eOooame~bnocmblo ~tep noceu4euu.~ O~a.~ { ~ n (0~); n > 0 }. TeopeMa 2 MOaCeT 6Wn, miTepnpeTHpoeaHa caeRy~oumM o6paaoM. C aepo~rr- H0C'na0 1 noc~e~o~aTe~HOCTb { 11 n; n > 0 } HMeeT n0CTOaHHOe MHOXeCTBO qac- THqH~X rrpe~emon H qaCTOTa nocemeHHa oKpeCTHOCTefl qaCTHqHUX npe~e~oe onH- ct,maeTcx ~epa~m (2). B riexoTopux cayqaHX yTBepaC~eHHa TeopeM 1 H 2 MOaCHO KoM6mmpoeaTb, wro npHnO~T K ~pro~HqecKol~ TeopeMe ~aH HCXO~HOi~ cay~IaflHoa nocae~oeaTe~,HOCrH (He 06H3aTem, H0 C0CTOJCUel~ H3 He3aBHCHmaX c~yqam~ux BCnHqHH [21/. IIem~ ~aHHOfl CTaTbH ~ IIOny'-IHTb aI-ia.rlor TeOpel~ml 2 Raa nocae~osa- TeabHOCTH (1). Kaa ~TOrO Ha~ nOHa~O6HTCa ~onOaHHTe~Hoe onpe~eaeHHe. Onpe~e~aeHHe 4 o ~ I e s ~ o6paaoM pacnpocTpaHaeTcH Ha cay~a~ r ~ep { ~n; n > I } H { ~'n ; n > 1 }, c y ~ e m ~ KOTOpI, IX Ha K ~ map KoHeqma. O n p e ~ e n e a ~ e 5, Bo~pacma~ouca~ noc.aeOooame.~bnOr ~tep (oo3~to.,w.no r ne,mtax) 9~ouea~enmna nocneOoaame.~bnocmu cqumwou~ux :~tep noc.~eDoaamenbnoc- mu { Un; n > 1 } ~ IRd na.~aaemc.~ noc.neOooamenbnocmb~o noranbnbvr :~tep noceu~enu~ noc,~eSo~raenbnocmu { Un; n > 1 } . 2. M e I ~ noeemeH~m ~.nm noc.ae~oea're.rn, Hoer~ ~'repam,~. [Ipe,m'io.noacHM, wro 0 p ~ r Ha (1) raxoea , Wro : : la r (0; 1): V x , y e ~Zd: IIq,(x)-,p(y)U < a l l x - y l l . IIyc'rt, c.ay~altmae eexTopu { ~n; n > 0 } x,~esyr pa,,,nq3eaeaemta a6como'mo Henpe- pummae oTaocuTeanato Mep~a JIe6era c naoT~oCTaml {Pn; n > 1 } . 155N 0041-6053. Ysp. J~un. ~ylm., 1999, m. 51, lV TM 1 MEPbI IIOCEI.UEHH$I H 3PFOfl~IqECKA.q TEOPEMA ... 125 T e o p e ~ a 3. 17ycmb cyu4ecm~yem 5openeoc~a~ dpynx~u~ p: IR d --~ (0; +**), cgna ~mopoa 1) cyu4ecmayem cxoOau~aac~ r +00 nocneSo~amenbnocmb {an; n > 1 } ma- ~a~, ~mo V n > 1: < a . = inf ~Pn(X-----2): Ilxll-<a.t 0 < (p(x) 2 ) a . ~ O , n ~ , , ~ . - ~ , , , n ~ ; 3) Vm_> 1, r - O , 1 . . . . . m - l : ~O~km+r = +co; k = l 4) lim 9(x) = u0. Tozaa cyu#ecm6yem ~tnomecm6o I2 o noano~ 6eponmnocmu maroe, nmo ann an96ozo co ~ f)o nocaeaoaamenbnocmb (Xn((0); n > 0} u~leem a r.anecmse nocneSosameabnocmu n o ~ b n ~ x ~tep noceulenua nocneaoeamenbnOCmb { $n Cr ; n >- > 0 } , eae n $n = Z a j , n>. 1, d ~ = p ( x - u o ) d x . j = l ]7:oxasamee.,em6o. ]i:Ia Ka~K~toro m > 1 onpeae~a~ nocaeaoBaTeymHOCrS {xmn;n>m} c~e~y~mJ~M o6paaoM: xm = ~n + ~0(~._~ +~(~.-2 +-"+~(~.-m+, +~(0)))... ). (3) 1-IOCKOYIbKy c13yHKIDIJt ~0 y/~OBY[eTBop~eT yCIIOBHIO flHrtmHt~a c a < 1 H o rpaHn- qeHa, TO lira li--m ~Xn-xm[[ = O ( m o d P ) . IIocTpOHM BHaqaYle IIOCJIe~OBaTCJIr~HOCTb JIOKad'IbHrdX Mep IIOCeI.UOHH$1 ]]d'IYl IIO- cJIe~oBaTe/I~HOCTH {xm; n > m} rlpH qbtlKCHpOBaHHOM mo PaCCMOTprlM ~n~l r = 0, . . . . . m - 1 noc~e~oBaTC~bHOCTS {XTm+r; j--> 1}. CornacHo (3) wro noc~e~oBa- 1 TeJI3bHOCTb HC3aBHCHMI~IX Cd"lyqal~HM.X ~ICMeHTOB B ~ d C_.rle/]OBaTe.rlbHO, aI-Ia.tI0rI4xl - 2, {XTm+r; j > 1} Ha MHOYK~@ rIOJ"IHOl~ B~poffTHOCTH HM@L~r CBO~I~i IIO- HO Tr CJlC~tosaTeJmHOC'e,,~o Mep noccmemltt nocJle~toaaTeamHOCaa, cymd pacnpe~e~etmtt m~a ' ~ --,/ m ~(-1) r~Xjm+r ] , n > 1. (4) j = l r o r o ~rro6u nccze~oaaT~ acH~rrroTaqeczoe noeeaeHHe n0CSle/I0eaTeSlbHOCTH Mep (4), pace~oTpa~ OTae~H0e c~araeMoe p(xm) (-~) npH n -~**. CornaeHO on- ISSN 0041-6033. Yrp. 7,~am. ~'ype. o 1999, m. 51, lq ~ 1 126 A.A./IOPOFOBL~B m pacnpc~encm~e Tor~a ~Jm Ol"paHHqC~UX a. O603Haq~,[ qCpr V n X 2 - - ~ n " ~OCTaTOqHO 60.T[bmHX HOMepOB n RdA z ~n I J p.(y-x).vr(dx), R a A IR a A IIOCKOJ~Ky cornacHo ycnosmo l i ra 13 n = 0, To /1 - -~ oo P - limli~nll.._.. = + ~ . HO~TOMy npn m >_ 1 - ~im ~(~2) = p- ~m ~(L% +~(xY-~')) = u0. P r/---) m n ---) oo Bcae~c~He 3IOZanbHOR HHTerpHpyeMocTn 0t3yHKRHH p m orpaHnqeHmocTn A R a ~ x I-..> I p ( y - x ) d y A ~.I~L~eTC~ Henpep~asHO~. l-[O~XOMy P{~: ~'q - ~nl ~ - ~ ) * , , , - ' ~ A Cornacao neM~e IITTo.rlbIla ,,(,,;.+,)'-"c~)- ~ , . + , I , c , - ~ o ~ , . , , ~ - . j = l j = l a IIozToMy ananoram~o [1] AJ~a rlocJie/Iol3aTeJl~HOe'l~ {xTm+r; j >- 1} nocxteAoBaTe.rn,- r locrb Mep ~176 ~IB3IHCTC,~ rIOC.TICKOBaTC.TIbHOCTbIO 3IOKa~bHMX ~tep IIOCCIKCHH,q Ha MHO~KeCTBr m - I noJmolt~epoaT~OCTn ~r" HyCTb CO e A ~r, U1 n U 2 --HenycT~e 0TKpm'ue r=0 o]"pa.B~.~c3-nR,zr MHo~e~rsa TaKHC, qTO ~ a l t ~ e U I , ~ S U 2. TorRa V s > O 3 n o : Vn>-no: n m - l Z ~.,(~7(=)) = Z Z " = Zu2(x~ ()) >_ kffi I rffiO k=r(n~dm),k<n m - I r=O kmr(modm),k~n n kffil lSYOl 0041.6053, Yicp. ,~am. ~.'ypn., ] 999. m. $2 , N e J MEPH FIOCE1J2EHH~I H 3PFO/~'IqECK_A~I TEOPEMA ... 127 CJqe/IOBaTeorlbHO, Ha MHO~KeCTBe rlO.rlHOR BepOaTHOCTH noc~e/IOBaTe~bHOCT/, Mep { 7n c ; n > 1 } aB~SeTcs noc~Ie~OBaTe~bHOCTb~O ZIOKa.m, HraX Mep noc e tue rms ~ s {x;: n_> 1}. o a aTo c a TeM, ~['0 U n- ! ---- 0 <mode m--)~o n -.) *o aHa.qoFrlqHO TOMy, KRK 9"1"O ~odIaJIOCb B [ 1 ]. TeopeMa KoKa3aHa. C~eacmoue (~or, a~bua~ 9pzo~u,~ecr, a ~ meope~a) . Hpu obmo/menuu yc/toout~ m e o p e ~ 3 an~ nlo6o~ uenpepbtono~ dpynxt~uu f : ~ d ~ ~ c Ko~natcmt~bt~t tuTcu- mene~t cnpaoeDnuoo paoencmeo It n|im ~ 1 ~ f ( x t ) = f f ( y ) f f (dy ) ( m o d e ) . Tn k = I ]~d 3a~te~tattue. YcaoBrIe cymecTBOBaHHS rrpe~eaa lira = u 0 MOaCHO 3mae- ] x I ~ + - - HHTb COBMeCTHIdM yc.rloBrleM Ha ~0 H rlOCYleaoBaTeabHOCTb {~n ; n -> 1 }, r apaaT~- p y ~ t t ~ M c~a6y~o CXO~nUOCTb Mep {vm; n > 1} K n p e z e a y , He 3aBacameMy O T m n M e ~ m e ~ J KOMnaKTmalt HOCHTe.qb. KpoMe TOI'O, rrpH p - 1 Ha/IO6HOCTo n yCJIOBHH 4 eCTeCTBeHHO oTrla~aeT i4 e r o MOJKHO 3aMeHaTb Tpe6OBaHHeM oFpaHHtIOHHOCTH qbyHKLmH ~0. l"~puYelep. [IyCTb { l~n; n ~_ 1 } ~ HOCJI~[OBaTeJIbHOCTb H~3aBHCHMhIX O~HHaKOBO pacnpe/leneHmaX rayccoBcKnx cJIyqal~HraX BeJIHqHH CO cpeRtlHM 0 H /mcncpcHcl t 1. Pacc~oTpHM rlOC~Ie/~OBaTeJIbHOCTI, rrrepatmtt Xn+ ~ = l s i n x n + n'qn, n>- l , x~ = O. 2 ~ I Z cJIy~a_qHHX BeJIHqHH ~n = nl]n, n > 1, BHIIO.rIHeHH yCYlOBH~I T e o p e ~ c c19yHK- 1 tmett p =-- 1, a n = n114, ~ n = ~ exp - n > 1. C~et~onaTen~HO, n o c ~ e - ~onareamHocr~ wrepatmla { x n; n > 1 } Ha MHOaCecTBe n o n a o ~ BepoaTaOCm rn~eeT I n n KaqeCTBe JIOKa.rlbHIdX Mep nocemeHHa Hoc~e/IoBaTe~bHOCT~ ~ . - , n > 2, r / Ie X ~ Mepa JIe6era. B aaCTHOCTn, ~ana npoHsBO~aOl~ CI3HHHTHOI~ HenpepraBHO~ qbyH- KIIJ, IH f C Bepo~rraocT~O 1 cHpaBe/~HBO paBeHCTBO 1. 2. ]lopoeo6~ee A. A. HeKOTOptae xapaKTepHcTHKU noc~e~onaTe~buocTetl HTepatmit co c,,nyqatlHta- MH ~MyatenMaMa//YKp. MaT. ~typu. -- 1996.- 48, N ~ 8. - C . 1047 - 1063. Dorogovtsev A. A., Denisievskii N. A. Path-wise behavior of stationary sequences // Theor. Stochast. Processes. - 1996. - 2 (I8), H ~ 3-4. - P. 17-26. rIo,nyqeuo 22.05.97 ISSN 0041-6053. Yr, p. ~uam, ~v. pn., 1999, m. 51, bl ~ 1
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spelling umjimathkievua-article-45902020-03-18T21:09:14Z Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations Меры посещения и эргодическая теорема для последовательности итераций со случайными возмущениями Dorogovtsev, A. A. Дороговцев, А. А. Дороговцев, А. А. By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem. За допомогою локальних мір перебування описано граничну поведінку послідовності ітерацій з випадковими не однаково розподіленими збуреннями. Як наслідок отримано варіант локальної ергодичної теореми. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4590 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 123–127 Український математичний журнал; Том 51 № 1 (1999); 123–127 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4590/5884 https://umj.imath.kiev.ua/index.php/umj/article/view/4590/5885 Copyright (c) 1999 Dorogovtsev A. A.
spellingShingle Dorogovtsev, A. A.
Дороговцев, А. А.
Дороговцев, А. А.
Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title_alt Меры посещения и эргодическая теорема для последовательности итераций со случайными возмущениями
title_full Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title_fullStr Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title_full_unstemmed Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title_short Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
title_sort visiting measures and an ergodic theorem for a sequence of iterations with random perturbations
url https://umj.imath.kiev.ua/index.php/umj/article/view/4590
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