Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solut...

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Datum:	2010
Hauptverfasser:	Bondarev, B. V., Kozyr', S. M., Бондарев, Б. В., Козырь, С. М.
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Sprache:	Russisch Englisch
Veröffentlicht:	Institute of Mathematics, NAS of Ukraine 2010
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Bondarev, B. V. Kozyr', S. M. Бондарев, Б. В. Козырь, С. М. Бондарев, Б. В. Козырь, С. М.
author_facet	Bondarev, B. V. Kozyr', S. M. Бондарев, Б. В. Козырь, С. М. Бондарев, Б. В. Козырь, С. М.
author_sort	Bondarev, B. V.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T19:40:12Z
description	We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space $C[0, T]$. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation.
first_indexed	2026-03-24T02:32:33Z
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fulltext	UDK 519.21 B. V. Bondarev, S. M. Koz¥r\ (Donec. nac. un-t) PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ SEMEJSTVA YNTEHRAL\|NÁX FUNKCYONALOV OT REÍENYQ DYFFERENCYAL\|NOHO URAVNENYQ S PERYODYÇESKYMY KO∏FFYCYENTAMY S SEMEJSTVOM VYNEROVSKYX PROCESSOV. NEKOTORÁE PRYLOÛENYQ. I We prove that a bounded 1-periodic function of a solution of time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the uniform strong intermixing condition. We establish an estimate for the rate of approach with respect to the probability in C T0,[ ] metric of some normed integral functional of a solution of ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of the Wiener processes. As an example, we consider an ordinary differential equation disturbed by a rapidly oscillating centered process, which is a 1-periodic function of a solution of time-homogeneous stochastic differential equation with 1-periodic coefficients. An estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation is established. Dovedeno, wo obmeΩena 1-periodyçna funkciq vid rozv’qzku odnoridnoho za çasom dyfuzijnoho rivnqnnq z 1-periodyçnymy koefici[ntamy utvorg[ proces, wo zadovol\nq[ umovu rivnomirnoho syl\noho peremißuvannq. Vstanovleno ocinku ßvydkosti zblyΩennq za jmovirnistg v metryci prostoru C T0,[ ] deqkoho normovanoho intehral\noho funkcionala vid rozv’qzku zvyçajnoho odnoridnoho za çasom stoxastyçnoho dyferencial\noho rivnqnnq z 1-periodyçnymy koefici[nta- my z sim’[g vinerovyx procesiv. Qk pryklad, rozhlqnuto zvyçajne dyferencial\ne rivnqnnq, zburene ßvydkooscylggçym centrovanym procesom, qkyj [ 1-periodyçnog funkci[g vid rozv’qzku odnoridnoho za çasom stoxastyçnoho dyferencial\noho rivnqnnq z 1-periodyçnymy koefici[ntamy. Vstanovleno ocinku ßvydkosti zblyΩennq rozv’qzku takoho rivnqnnq z rozv’qz- kom vidpovidnoho stoxastyçnoho rivnqnnq Ito. 1. Vvedenye. Yzvestno (sm., naprymer, [1 – 14]), çto pry nekotor¥x uslovyqx centryrovann¥j sluçajn¥j process W s dst t ε ε ε ξ= ∫ ( ) / 0 , t T∈[ ]0, , sblyΩaetsq (v tom yly ynom sm¥sle) pry ε → 0 s nekotor¥m semejstvom vyne- rovskyx processov σ εW t( ) , t T∈[ ]0, , hde σ ξ ξ=         ∞ ∫2 0 0 1 2 M ( ) ( ) / t dt . V rabotax [1, 5 – 8, 14] pryveden¥ dostatoçn¥e uslovyq dlq toho, çtob¥ sluçaj- n¥j process Wt ε slabo sxodylsq pry ε → 0 k vynerovskomu processu σW t( ) , t T∈[ ]0, . V rabotax [2 – 4, 9 – 11] ustanovlen¥ dostatoçn¥e uslovyq dlq toho, çtob¥ semejstvo sluçajn¥x processov Wt ε pry dostatoçno mal¥x ε > 0 b¥lo „blyzko” po veroqtnosty v metryke prostranstva C T0,[ ] k vynerovskomu se- mejstvu σ εW t( ) , t T∈[ ]0, , a ymenno ustanovlen¥ neravenstva vyda P sup 0≤ ≤ { t T tW ε – σ δε εW t( ) > } < γ ε , hde funkcyy δε → 0, γ ε → 0 pry ε → 0 © B. V. BONDAREV, S. M. KOZÁR\|, 2010 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 733 734 B. V. BONDAREV, S. M. KOZÁR\| zapysan¥ v qvnom vyde. V rabotax [12, 13] pryveden¥ dostatoçn¥e uslovyq dlq toho, çtob¥ semejstvo sluçajn¥x processov Wt ε pry dostatoçno mal¥x ε > 0 b¥lo „blyzko” k vynerovskomu semejstvu σ εW t( ) , t T∈[ ]0, , v sm¥sle sup 0≤ ≤t T tWM ε – σ εW t( ) 2 ≤ γ ε , hde funkcyq γ ε → 0 pry ε → 0 zapysana v qv- nom vyde. V rabotax [2 – 4, 9 – 13] na process ξ( )t , t ≥ 0, lybo naklad¥valos\ uslovye ravnomernoho syl\noho peremeßyvanyq (kotoroe naz¥vagt takΩe ϕ- peremeßyvanyem [7, s. 165] yly peremeßyvanyem „po Ybrahymovu” [15, s. 11]), lybo yspol\zovalsq process ξ( )t , t ≥ 0, konkretnoho vyda [1]. V nastoqwej ra- bote v kaçestve processa ξ( )t , t ≥ 0, rassmatryvaetsq reßenye odnorodnoho po vremeny stoxastyçeskoho dyfferencyal\noho uravnenyq s peryodyçeskymy koπffycyentamy, a ymenno, reßenye uravnenyq ξ( )t = ξ( )0 + α ξ( )s ds t ( )∫ 0 + β ξ( ) ( )s dW s t ( )∫ 0 , (1) hde W s( ) — standartn¥j vynerovskyj process, ξ( )0 — v obwem sluçae slu- çajnoe (ne zavysqwee ot W s( ) ) naçal\noe uslovye, pryçem v sylu toho, çto bu- det yzuçat\sq process f tξ( )( ) , hde f x( ) — 1-peryodyçeskaq funkcyq, dlq dal\nejßyx v¥kladok bez narußenyq obwnosty dostatoçno rassmatryvat\ slu- çaj ξ( )0 ∈ 0 1,[ ) . V dal\nejßem budem sçytat\, çto vsehda v¥polnqgtsq sledugwye uslovyq: 1) 1-peryodyçeskye koπffycyent¥ α( )x , β( )x ymegt proyzvodn¥e pervoho porqdka ′α ( )x , ′β ( ),x udovletvorqgwye uslovyg Hel\dera; 2) funkcyy α( )x , β( )x , ′α ( )x , ′β ( ),x takov¥, çto α α( ) ( )x x= + 1 , β β( ) ( )x x= + 1 , α( )x ≤ K < + ∞, 0 < λ ≤ β2( )x ≤ K < + ∞, ′α ( )x + ′β ( )x ≤ K < + ∞. V kaçestve Wt ε budut yssledovan¥ process¥ W f s f s dst t ε ε ε ξ ξ= ( ) − ( )[ ]∫ ( ) ( ) / M 0 , W f s f dst t ε ε ε ξ= ( ) −[ ]∫ ( ) / 0 . Zdes\ y dalee f x( ) — dvaΩd¥ neprer¥vno dyfferencyruemaq ohranyçennaq f x K( ) ≤ < +∞( ) 1-peryodyçeskaq funkcyq, f — ee srednee, v¥çyslennoe po nekotoromu ynvaryantnomu raspredelenyg (sm. nyΩe formulu (12)), f tξ( )( ) — stacyonarn¥j markovskyj process. Dlq ocenky skorosty sblyΩenyq Wt ε s sootvetstvugwym obrazom postroenn¥m semejstvom vynerovskyx processov σ εW t( ) yspol\zuetsq metod odnoho veroqtnostnoho prostranstva A.FV.FSkoro- xoda, a mera uklonenyq v prostranstve traektoryj — metryka prostranstva C T0,[ ] . Pry postroenyy sootvetstvugwyx ocenok yspol\zuetsq martynhal\- naq approksymacyq D.FO.FÇykyna [6]. Postroennoe martynhal\noe pryblyΩenye pozvolylo yspol\zovat\ pry dokazatel\stve osnovnoho rezul\tata yzvestn¥e ocenky dlq uklonenyq za uroven\ stoxastyçeskyx yntehralov v metryke C T0,[ ] . ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 735 2. Ocenka skorosty sblyΩenyq normyrovann¥x yntehralov ot dyffu- zyonn¥x processov s peryodyçeskymy koπffycyentamy s semejstvom ne- prer¥vn¥x martynhalov. Pust\ ρ( )x — reßenye peryodyçeskoj zadaçy L*ρ = 1 2 2 2 2 d x x dx ρ β( ) ( )( ) – d x x dx ρ α( ) ( )( ) = 0, ρ ρ( ) ( )x x= + 1 , (2) odnoznaçno opredelennoe uslovyem normyrovky ρ( )x dx =∫ 1 0 1 . (3) Najdem reßenye zadaçy (2). Pust\ (sm. [14]) funkcyq ϑ α β ( ) exp ( ) ( ) x y y dy x = −         ∫ 2 2 0 . Netrudno zametyt\, çto pry x ∈[ ]0 1, dlq ϑ( )x spravedlyv¥ ocenky 0 < exp −{ }2K λ = C−1 ≤ ϑ( )x ≤ C = exp 2K λ{ } < + ∞. (4) Rassmotrym (sm. [14]) funkcyg G x0( ) = 2 1 1 12 1 0+( ) +   ∫∫ϑ β ϑ ϑ ϑ ϑ ( ) ( ) ( ) ( ) ( ) ( ) x x y dy y dy x x       . (5) Netrudno zametyt\, çto vvedennaq funkcyq G x0( ) qvlqetsq 1-peryodyçeskoj. Dejstvytel\no, v sylu toho, çto ϑ( )x + 1 = ϑ ϑ( ) ( )x 1 , ymeem G x0 1( )+ = 2 1 1 1 12 1 1 +( ) − + ϑ ϑ β ϑ ϑ ϑ ϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) x x y dy y dy x ∫∫∫ +       0 1x = = 2 1 1 1 12 00+( ) + − ∫ϑ β ϑ ϑ ϑ ϑ ( ) ( ) ( ) ( ) ( ( ) ) ( ) x x y dy y dy x11 ∫         = G x0( ) . Netrudno ubedyt\sq takΩe v tom, çto funkcyq G x0( ) udovletvorqet urav- nenyg 1 2 2 0β ( ) ( )x G x( )′ = G x x0( ) ( )α + 1 1 1 1 1 +( ) −[ ] ϑ ϑ ( ) ( ) . V sylu suwestvovanyq proyzvodn¥x ′α ( )x , ′β ( )x y uslovyq 0 < λ ≤ β2( )x sleduet dyfferencyruemost\ v¥raΩenyj β2 0( ) ( )x G x( )′ y G x x0( ) ( )α . Takym obrazom, ρ( ) ( ) ( )x G x G y dy=         ∫ − 0 0 0 1 1 (6) qvlqetsq klassyçeskym reßenyem zadaçy (2) , udovletvorqgwym (3). PokaΩem, çto ne narußaq obwnosty moΩno rassmatryvat\ lyß\ sluçaj, kohda reßenye (1) startuet yz toçky � x , � x ∈[ )0 1, . Dejstvytel\no, pust\ ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 736 B. V. BONDAREV, S. M. KOZÁR\| x k k∈ +[ ), 1 , k Z∈ , Z = 0 1 1 2 2, , , , ,+ − + − …{ } , a ξ x t( ) = x + α ξ x t s ds( )( )∫ 0 + β ξ x t s dW s( ) ( )( )∫ 0 , yly ξ x t( ) = � x + k + α ξ x t s ds( )( )∫ 0 + β ξ x t s dW s( ) ( )( )∫ 0 , yly ξ x t k( ) − = � x + α ξ x t s k k ds( ) − +( )∫ 0 + β ξ x t s k k dW s( ) ( )− +( )∫ 0 . Yz posledneho v sylu peryodyçnosty koπffycyentov α( )x , β( )x poluçaem η( )t = � x + α η β η( ) ( ) ( )s ds s dW s t t ( ) + ( )∫ ∫ 0 0 , hde η( )t = ξ x t+1( ) – k . V sylu edynstvennosty reßenyq posledneho uravnenyq η ξ( ) ( )t tx= � , ξ ξx xt k t( ) ( )= + � , a v sylu peryodyçnosty f x( ) ymeem f txξ ( )( ) = f txξ � ( )( ) . Postroym process � �ξ x t( ) , prynymagwyj svoy znaçenyq v polose ( , )x t{ : 0 1,[ ) × 0, +∞[ )} , plotnost\ veroqtnosty perexoda kotoroho yz toçky � x ∈[ )0 1, v toçku � z ∈[ )0 1, za vremq t > 0 podsçyt¥vaetsq po formule [16, s. 372] � � � � � p x t z p x t z k k Z , , , ,( ) = +( ) ∈ ∑ . Zdes\ p x t z � , ,( ) — plotnost\ veroqtnosty perexoda yz toçky � x ∈[ )0 1, v toçku z ∈ −∞ +∞( ), za vremq t > 0 processa ξ � x t( ) , startugweho v nulevoj moment vremeny yz toçky � x ∈[ )0 1, . Otmetym, çto uslovyq 1, 2 harantyrugt suwestvo- vanye plotnosty raspredelenyq p y t z( , , ) � [8, s. 30]. Process � �ξ x t( ) ymeet πr- hodyçeskoe raspredelenye s plotnost\g ρ( )x . Takym obrazom, esly v kaçestve naçal\noho uslovyq v (1) vzqt\ ξ( )0 — ne zavysqwug ot W t( ) sluçajnug ve- lyçynu, kotoraq ymeet plotnost\ raspredelenyq ρ( )x , to postroenn¥j v¥ße process � ξ( )t s takym naçal\n¥m uslovyem budet stacyonarn¥m markovskym processom, a znaçyt process f t � ξ( )( ) , a s nym y process �η( )t = f t f � ξ( )( ) − = f t f t � � ξ ξ( ) ( )( ) − ( )M budut stacyonarn¥my markovskymy processamy, a process � Wt ε . = 1 ε ξ ε f t f �         −    = 1 ε ξ ε ξ ε f t f t� �        −             M budet stacyonarn¥m „fyzyçeskym” bel¥m ßumom. ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 737 Netrudno takΩe zametyt\, çto dlq 1-peryodyçeskoj funkcyy f x( ) spra- vedlyvo sledugwee [16, s. 372]: M f txξ �( )( ) = f y p x t y dy( ) ( , , ) −∞ +∞ ∫ � = f y p x t y dy k k k ( ) ( , , ) + =−∞ +∞ ∫∑ 1 � = = f y p x t y k dy k ( ) ( , , ) � � � � 0 1 ∫ ∑ + =−∞ +∞ = f y p x t y dy f tx( ) ( , , ) ( ) � � � � � � � 0 1 ∫ = ( )M ξ . Pust\ 0 < t1 < t2 < … < tn < + ∞, tohda ϕξ ( , , , )z z zn1 2 … = M exp ( )iz f tk k k n ξ( )   = ∑ 1 = = exp ( ) ( ) , ,iz f x x p x t xk k k n = ∑∫     ( ) 00 1 0 0 1 1ρ � � −−∞ +∞ ∫ … … p x t t x dx dx dx dxn n n n n n− − −∞ +∞ −−( ) …∫ 1 1 1 1 0, , � = = exp ( ) ( ) , ,iz f x x p x t xk k k n = ∑∫     ( ) 00 1 0 0 1 1ρ � � kk k k + =−∞ +∞ ∫∑ 1 … … p x t t x dx dx dx dn n n n n k k k n− − + =−∞ +∞ −−( ) …∫∑ 1 1 1 1 1, , �� x0 = = exp ( ) ( ) , ,iz f x x p x t xk k k n � � � � = ∑∫    00 1 0 0 1 1ρ ++( )∫∑ =−∞ +∞ k k 1 0 1 1 … … p x t t x k dx dxn n n n n n k n n − − =−∞ +∞ −− +( )∫∑ 1 1 0 1 1, , � � � …… dx dx � � 1 0 = = exp ( ) ( ) , ,iz f x x p x t xk k k n � � � � � = ∑∫    00 1 0 0 1ρ 11 0 1 ( )∫ … … � � � � � � p x t t x dx dx dx dxn n n n n n− − −−( ) …∫ 1 1 0 1 1 1 0, , = = M exp ( )iz f tk k k n � ξ( )   = ∑ 1 = ϕξ � ( , , , )z z zn1 2 … , t.Fe. xarakterystyçeskye funkcyy koneçnomern¥x raspredelenyj sovpadagt, a znaçyt, koneçnomern¥e raspredelenyq u processov �η( )t = f t f � ξ( )( ) − = f t f t � � ξ ξ( ) ( )( ) − ( )M y η( )t = f t fξ( )( ) − = f t f tξ ξ( ) ( )( ) − ( )M ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 738 B. V. BONDAREV, S. M. KOZÁR\| odny y te Ωe. Otsgda sleduet, çto process Wt ε . = 1 ε ξ εf t f/( )( ) −[ ] = 1 ε ξ ε ξ εf t f t/ /( )( ) − ( )( )[ ]M takΩe budet stacyonarn¥m „fyzyçeskym” bel¥m ßumom. Yzvestno [16, s. 370, 373], çto esly α( )x , β( )x — 1-peryodyçeskye koπffy- cyent¥ uravnenyq (1) — ymegt proyzvodn¥e pervoho porqdka ′α ( )x , ′β ( )x , udovletvorqgwye uslovyg Hel\dera, y v¥polneno uslovye 0 < λ ≤ β2( )x ≤ K < < + ∞, to dlq lgboj ohranyçennoj 1-peryodyçeskoj funkcyy f x( ) spravedly- va ocenka sup ( ) x xf t fM ξ( ) − ≤ c f x t x sup ( ) exp −{ }γ , (7) hde postoqnn¥e c > 0, γ > 0 opredelqgtsq çerez koπffycyent¥ uravnenyq (1), ξ x t( ) — reßenye uravnenyq ξ x t( ) = x + α ξ x t s ds( )( )∫ 0 + β ξ x t s dW s( ) ( )( )∫ 0 , postoqnnaq f podsçyt¥vaetsq po formule f f x x dx= ∫ ( ) ( )ρ 0 1 , (8) ρ( )x opredeleno v (6). Pust\ ς y t( ) = f txξ ( )( ) , t ≥ 0, — process s naçal\n¥m uslovyem y f x= ( ) , I yA( ) — yndykator proyzvol\noho mnoΩestva A yz oblasty znaçenyj ς y t( ) . Tohda I t I f t g tA y A x A xς ξ ξ( ) ( ( ) ( )( ) = ( ) = ( ) , hde g xA( ) = I f xA ( )( ) — takΩe ohranyçennaq 1-peryodyçeskaq funkcyq, dlq kotoroj spravedlyvo sootnoßenye (7), kotoroe v dannom sluçae prynymaet vyd sup ( ) ( ) ( ) ,y A A y AI t I f x x dxM ς ρ( ) − ( )∫ 0 1 ≤ c texp −{ }γ . Esly π( )A — mera, opredelennaq sootnoßenyem π ρ( ) ( ) : ( ) A x dx x f x A = ∈{ } ∫ , to ymeem sup , , ( ) ,y A y t A AP{ } − π = sup ( ) ( ) ( ) ,y A A y AI t I f x x dxM ς ρ( ) − ( )∫ 0 1 ≤ ≤ c texp −{ }γ , t.Fe. v¥polnqetsq uslovye (19.1.7) yz [17], dostatoçnoe dlq toho, çtob¥ process f tξ( )( ) (naçal\noe uslovye raspredeleno s plotnost\g ρ( )x ) udovletvorql us- lovyg ravnomernoho syl\noho peremeßyvanyq s koπffycyentom peremeßyva- ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 739 nyq, ne prev¥ßagwym 2c texp −{ }γ . Takym obrazom, klass sluçajn¥x proces- sov, kotor¥j stroytsq kak superpozycyq 1-peryodyçeskoj funkcyy ot reßenyq dyffuzyonnoho uravnenyq s 1-peryodyçeskymy koπffycyentamy, budet pere- meßyvat\sq „po Ybrahymovu”. Kak pokaz¥vagt pryvedenn¥e nyΩe prymer¥, πto ves\ma ßyrokyj klass. Rassmotrym funkcyg U x f t f dtx( ) ( )= − ( ) −[ ] +∞ ∫ M ξ 0 , (9) hde ξ x t( ) = x + α ξ x t s ds( )( )∫ 0 + β ξ x t s dW s( ) ( )( )∫ 0 , (10) a f x( ) — dvaΩd¥ neprer¥vno dyfferencyruemaq ohranyçennaq f x( )( ≤ K < < +∞) 1-peryodyçeskaq funkcyq (v sylu uslovyq (7) yntehral v (9) opredelen). PokaΩem, çto U x( ) — 1-peryodyçeskaq funkcyq, kotoraq qvlqetsq reßenyem uravnenyq β2 2 22 ( ) ( ) x d U dx x + α( ) ( )x dU dx x = f x f( ) −[ ] . (11) Ustanovym peryodyçnost\ funkcyy U x( ) . Pust\ x ∈ k k, +[ )1 , k Z∈ , Z — mnoΩestvo cel¥x çysel, a ξ x t( ) — reßenye uravnenyq (10). Narqdu s (10) ras- smotrym uravnenye ξ x t+1( ) = x + 1 + α ξ x t s ds+( )∫ 1 0 ( ) + β ξ x t s dW s+( )∫ 1 0 ( ) ( ) . Yz posledneho v sylu peryodyçnosty koπffycyentov α( )x , β( )x ymeem η( )t = x + α η( )s ds t ( )∫ 0 + β η( ) ( )s dW s t ( )∫ 0 , (12) hde η( )t = ξ x t+1( ) – 1. V sylu edynstvennosty reßenyq (10) yz (12) sleduet η ξ( ) ( )t tx= , ξ ξx xt t+ = +1 1( ) ( ) , a v sylu peryodyçnosty f x( ) ymeem f txξ +( )1( ) = f txξ ( )( ) , otkuda v sylu predstavlenyq (9) sleduet peryodyçnost\ U x( ) . Dalee, pust\ V t x f t fx( , ) ( )= − ( ) −[ ]M ξ , tohda (sm. [18]) funkcyq V t x( , ) qvlqetsq reßenyem zadaçy β2 2 22 ( ) ( , ) x V x t x ∂ ∂ + α( ) ( , )x V x t x ∂ ∂ = ∂ ∂ V t t x( , ) , (13) V x f x f( , ) ( )0 = − −[ ] . Yntehryruq obe çasty (13) po t v predelax ot 0 do T y perexodq k predelu pry T → + ∞, s uçetom ocenky (7) ymeem ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 740 B. V. BONDAREV, S. M. KOZÁR\| β2 2 22 ( ) ( ) x U x x ∂ ∂ + α( ) ( )x U x x ∂ ∂ = lim T T V t t x dt →+∞ ∂ ∂∫ ( , ) 0 = = − →+∞ lim T V T x( , ) + f x f( ) −[ ] = − −[ ] →+∞ lim M T xf T f( ( ))ξ + + f x f( ) −[ ] = f x f( ) −[ ] , t.Fe. (11) ymeet mesto. Yz (9) vsledstvye (7) sleduet ohranyçennost\ U x( ) . Dejstvytel\no, U x( ) ≤ M f t f dtx( ( ))ξ −[ ] +∞ ∫ 0 ≤ Kc t dtexp −{ } +∞ ∫ γ 0 = Kc γ . (14) Oboznaçym ψ( )x = dU dx , tohda yz (11) sleduet, çto ψ( )x qvlqetsq reßenyem zadaçy d dx x x x ψ α β ψ+ 2 2 ( ) ( ) ( ) = 2 2 f x f x ( ) ( ) −[ ] β , ψ ψ( ) ( )x x+ =1 . (15) Netrudno ubedyt\sq v tom, çto reßenyem zadaçy (15) budet funkcyq ψ( )x = − −[ ]        − ∫ϑ ϑ β 1 2 1 2 ( ) ( ) ( ) ( ) x y f y f y dy x . (16) V sylu peryodyçnosty ψ( )x y (4) ymeem ocenku ψ λ ψ( )x C K D≤ = 4 2 . (17) Pust\ U x( ) — reßenye zadaçy (11). Tohda, prymenqq formulu Yto k processu U txξ ( )( ) , poluçaem dU txξ ( )( ) = LU t dtxξ ( )( ) + β ξ ψ ξx xt t dW t( ) ( ) ( )( ) ( ) . Otsgda v sylu (11) sleduet dU txξ ( )( ) = f t f dtxξ ( )( ) −[ ] + β ξ ψ ξx xt t dW t( ) ( ) ( )( ) ( ) . Yntehryruq poslednee uravnenye v predelax ot 0 do t /ε , ymeem ε ξ ε f s f dsx t ( ( )) / −[ ]∫ 0 = − ( ) −[ ]ε ξ εU t U xx ( / ) ( ) – – ε β ξ ψ ξ ε x x t s s dW s( ) ( ) ( ) / ( ) ( )∫ 0 . (18) Yz (18) sleduet ravenstvo ε ξ ξ ε f s f s dsx x t ( ) ( ) / ( ) − ( )[ ]∫ M 0 = − ( ) −[ ]ε ξ εU t U xx ( / ) ( ) – – ε β ξ ψ ξ ε x x t s s dW s( ) ( ) ( ) / ( ) ( )∫ 0 – ε ξ ε M f s f dsx t ( ) / ( ) −[ ]∫ 0 . ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 741 Netrudno zametyt\, çto predstavlenye (18) — πto predstavlenye D.FO.FÇyky- naF[6]: v dannom sluçae ρε t = ε ξ εU x U tx( ) ( / )− ( )[ ], µε t = − ( ) ( )∫ε β ξ ξε x t xs dU s dx dW s( ) ( ) ( ) / 0 . Takym obrazom, spravedlyva sledugwaq teorema. Teorema 1. Pust\ dyffuzyonn¥j process zadan kak reßenye uravnenyq (1) y v¥polnen¥ uslovyq 1, 2. Tohda dlq lgboj 1-peryodyçeskoj dvaΩd¥ nepre- r¥vno dyfferencyruemoj funkcyy f x( ) f x K( ) ≤ < +∞( ) y lgboho 0 < ε ≤ 1 spravedlyva ocenka sup ( ) ( ) ( ) 0 ≤ ≤ ( ) −[ ] + ( ) ( ) t T x x xf s f ds s s dWε ξ ε β ξ ψ ξ (( ) // s tt 00 εε ∫∫ ≤ ε γ 2 Kc , (19) hde postoqnn¥e c > 0, γ > 0 vzqt¥ yz ocenky (7), funkcyq ψ( )x zadaetsq formuloj (16), a ρ( )x — formuloj (6), v kotoroj G x0( ) zadano v (5). Zameçanye 1. Vospol\zovavßys\ ocenkamy (19) y (7), netrudno poluçyt\ ocenku sup ( ) ( ) ( ) 0 ≤ ≤ ( ) − ( )[ ] + ( ) t T x x xf s f s ds sε ξ ξ ε β ξ ψ ξM xx tt s dW s( ) ( ) // ( )∫∫ 00 εε ≤ ≤ ε γ 2 Kc + ε ξ ε Mf s f dsx T ( ) / ( ) −∫ 0 ≤ ε γ 2 Kc + + Kc t dt T ε γ ε exp / −{ }∫ 0 ≤ ε γ 3 Kc . 3. Ocenka skorosty sblyΩenyq semejstva normyrovann¥x stoxastyçes- kyx yntehralov ot dyffuzyonn¥x processov s peryodyçeskymy koπffy- cyentamy s semejstvom vynerovskyx processov. Osnovn¥m rezul\tatom πto- ho punkta qvlqetsq sledugwaq teorema. Teorema 2. Pust\ ρ( )x — plotnost\ raspredelenyq, kotoraq zadaetsq formuloj (6), dyffuzyonn¥j process zadan kak reßenye uravnenyq (1) s na- çal\n¥m znaçenyem ξ( )0 , ne zavysqwym ot W t( ) , t ≥ 0 ymegwym raspredele- nye s plotnost\g ρ( )x , v¥polnen¥ uslovyq 1, 2, y ψ β ρ2 2 0 1 ( ) ( ) ( )x x x dx∫ = σ2 0> . Tohda dlq lgboho 0 < σ < 1 4/ spravedlyva ocenka P sup ( ) ( ) ˜ ( ) / 0 0≤ ≤ ( ) − ( )[ ] + >         ≤∫ t T t f s Mf s ds W tε ξ ξ σ δ γε ε ε ε, (20) hde �W tε ( ) — nekotoroe semejstvo standartn¥x vynerovskyx processov, δ ε ε ε γε δ γ ε δ = + +( ) +− − 6 12 41 4 T ce Kc/ / , ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 742 B. V. BONDAREV, S. M. KOZÁR\| γ ε = 20 1 32 8 2 3 1 4 T K Kε λ λ ε δ +     − { }       − exp exp /   + + 4 2 2 1 2 2 2 2exp /−      − σ ε δ TD + +F exp / / −       + +         − 1 256 1 1 2 2 11 2 1 2 K T ε εδ   + T ce ε γ ε δ +     − − 1 12 1 4/ / , (21) postoqnn¥e c > 0, γ > 0 vzqt¥ yz ocenky (7), a D2 = 16 4 2exp K cK K λ λγ { } + 4 2 2 2 K c γ . Dokazatel\stvo. Budem sledovat\ sxeme rassuΩdenyj yz [3, 4]. Snaçala otmetym, çto spravedlyva ocenka P sup ( ) ( ) ( ) t t h t s s dW s x ≤ ≤ + ( ) ( ) >      ∫ τ τ ε ψ ξ β ξ   ≤ 2 2 2 2exp −         x hKDε ψ , (22) hde postoqnnaq Dψ opredelena v (17). V spravedlyvosty (22) netrudno ubedyt\sq, esly vospol\zovat\sq πksponen- cyal\noj ocenkoj dlq supremuma neprer¥vnoho martynhala µ( )t [19, s. 173], zametyv, çto v dannom sluçae ymeet mesto ocenka εµ εµ, t t h+ ≤ ε ψhKD2 . Pust\ ηε ( )t = ε ψ ξ β ξ ε ( ) ( ) ( ) / s s dW s t ( ) ( )∫ 0 , ςε ( )t = ε ξ ξ ε f s f s ds t ( ) ( ) / ( ) − ( )[ ]∫ M 0 = ε ξ ε f s f ds t ( ) / ( ) −[ ]∫ 0 . V sylu toho, çto ς ε ξ ξ ε ηε ε( ) ( ) ( )t U U t t= ( ) −             −0 , s uçetom (7) y (14) pry τ ≥ t ymeem ocenku ς τ ςε ε( ) ( )− t ≤ ε γ 2Kc + η τ ηε ε( ) ( )− t . S uçetom posledneho y ocenky (22) netrudno ubedyt\sq v tom, çto ymeet mesto neravenstvo P sup ( ) ( ) t t h t x Kc ≤ ≤ + − > +      τ ε ες τ ς ε γ 2 ≤ 2 2 2 2exp −         x hKDε ψ . (23) ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 743 Pust\ lε = o 1 ε     → ∞, ε → 0, l T ε ε < , n = T lε ε     — celaq çast\ T lε ε , vremenn¥e otrezky bε , hε takye, çto T nl r ε ε ε= + , l b hε ε ε= + , hε = o l( )ε → ∞, ε → 0, — ynterval¥ dlq razbyenyq S.FN.FBernßtejna (sm., naprymer, [4]). Ydeq raz- byenyq Bernßtejna sostoyt v tom, çto summa sluçajn¥x velyçyn razbyvaetsq na dve çasty: na osnovnug summu (no uΩe so slabozavysym¥my slahaem¥my) y prenebreΩymug çast\, kotoraq po veroqtnosty budet stremyt\sq k nulg. Vve- dem oboznaçenyq νε k = ε ψ ξ β ξ ε ε ε ( ) ( ) ( )s s dW s kl b kl ( ) ( ) − ∫ , λε k = ε ψ ξ β ξ ε ε ( ) ( ) ( ) ( ) s s dW s k l kl ( ) ( ) − ∫ 1 , µε k = ε ψ ξ β ξ ε ε ε ( ) ( ) ( ) ( ) ( ) s s dW s k l k l h ( ) ( ) − − + ∫ 1 1 , αε k = ε ξ ξ ε ε ε f s f s ds k l k l h ( ) ( ) ( ) ( ) ( ) − ( )[ ] − − + ∫ M 1 1 , βε k = f s f s ds kl b kl ξ ξ ε ε ε ( ) ( )( ) − ( )[ ] − ∫ M , γ α ε βε ε ε k k k= + , k = 1, 2, … , n. Dalee, pust\ η1 , η2 , … — posledovatel\nost\ sluçajn¥x velyçyn, udovletvo- rqgwyx uslovyg Mηk = 0, Dηk < + ∞, 0 = t0 < t1 < … < tk−1 < tk < …F— toç- ky na vremennoj osy, η( )0 = 0, η( )k = ηii k =∑ 1 , k = 1, 2, … . Par¥ t kk , ( )η( ) , k = = 0, 1, … , budem naz¥vat\ uzlamy. Budem rassmatryvat\ sluçajn¥e stupençat¥e funkcyy vyda η( )t = η( )k − 1 , η( )0 = 0 pry t ∈F[ , )t tk k−1 , k = 1, 2, … . Pust\ s tε 1( ) — sluçajnaq stupençataq funkcyq s uzlamy k l i i k ε γε ε, = ∑   1 , k = 0, 1, … , n, = = ∑ 0 1 0 i , hde n = T lε ε     — celaq çast\ T lε ε . Tohda na vremennom otrezke k lε ε[ , (k + + 1) ε εl ) spravedlyv¥ ocenky sup ( ) ( ) ( )k l t k l t t ε ε ε ε ε ε ς ς ≤ ≤ + − 1 1 ≤ ε ς ς ε ε ε ε ε ε ε ε sup ( ) ( ) ( )k l t k l t k l ≤ ≤ + − 1 , (24) αε k ≤ ε ς ς ε ε ε ε ε ε ε ε ε ε sup ( ) ( ) ( ) ( )k l t k l h t k l − ≤ ≤ − + − − 1 1 1(( ) . Yspol\zuq pervug yz ocenok (24), s uçetom ocenky (23) poluçaem ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 744 B. V. BONDAREV, S. M. KOZÁR\| P sup ( ) ( ) ( )k l t k l t t x Kc ε ε ε ε ε ε ς ς ε γ≤ ≤ + − > +    1 1 1 2    ≤ 2 2 1 2 2exp −         x l KDε ε ψ . S uçetom πtoj ocenky ymeem P sup ( ) ( ) 0 1 1 2 ≤ ≤ − > +     t T t t x Kc ς ς ε γε ε ≤ ≤ − > ≤ ≤ ≤ ≤ + P max sup ( ) ( ) ( )0 1 1 1 k n k l t k l t t x ε ε ε ε ε ε ς ς ++       ε γ 2 Kc ≤ ≤ 2 1 2 1 2 2 T l x l KDε εε ε ψ +    −         exp . (25) Pust\ ςε 2( )t — sluçajnaq stupençataq funkcyq s uzlamy k l i i k ε ε βε ε, = ∑   1 , k = 0, … , n, = = ∑ 0 1 0 i . Tohda, yspol\zuq vtorug yz ocenok (24), s uçetom (23) poluçaem P sup ( ) ( ) 0 2 1 2 2 ≤ ≤ − > +      t T t t x Kc ς ς ε γε ε ≤ ≤ P max 1 1 2 2 ≤ ≤ = ∑ > +        k n i i k x Kcα ε γ ε ≤ P α ε γ ε i i n x Kc > +        = ∑ 2 1 2 ≤ ≤ 2 1 2 2 2 2 T l x h KDε εε ε ψ +    −         exp . (26) Yz (25) y (26) sleduet ocenka P sup ( ) ( ) 0 2 1 2 4 ≤ ≤ − > + +     t T t t x x Kc ς ς ε γε ε ≤ ≤ 2 1 2 1 2 2 T l x l KDε εε ε ψ +    −         exp + 2 1 2 2 2 2 T l x h KDε εε ε ψ +    −         exp . Yz posledneho pry x x1 2= ymeem ocenku P sup ( ) ( ) 0 2 12 4 ≤ ≤ − > +     t T t t x Kc ς ς ε γε ε ≤ 4 1 2 1 2 2 T l x l KDε εε ε ψ +    −         exp . (27) Otmetym, çto uzl¥ u lomanoj ςε 2( )t postroen¥ po summam slabozavysym¥x ve- lyçyn. V dal\nejßem nam ponadobytsq analoh lomanoj ςε 2( )t , no uΩe s uzla- my, postroenn¥my po summam nezavysym¥x velyçyn. Dejstvytel\no, uslovye (7) dostatoçno dlq toho, çtob¥ posledovatel\nost\ βε i{ } , i = 1, 2, … , peremeßyva- las\ „po Ybrahymovu”, pryçem v sylu toho, çto sluçajn¥e velyçyn¥ βε 1{ , … ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 745 … , βε n} otstoqt druh ot druha po vremeny ne menee çem na velyçynu hε , σ-al- hebra ℑ1 khε , poroΩdennaq velyçynamy βε i{ } , i = 1, … , k, y σ-alhebra, poroΩ- dennaq velyçynamy ℑ +kh h nh ε ε ε , takΩe otstoqt po vremeny druh ot druha ne menee çem na velyçynu hε . Tohda dlq koπffycyenta peremeßyvanyq „po Ybrahymo- vu” ϕ ε( )h spravedlyva ocenka [17, s. 467] ϕ γε ε( ) exph c h≤ −{ }2 . (28) Pust\ vektor � �β βε ε 1 , ,…{ }n sostoyt yz nezavysym¥x sluçajn¥x velyçyn �βε i , i = 1, … , n, kaΩdaq yz kotor¥x ymeet takoe Ωe raspredelenye, kak y βε i , i = = 1, … , n. Pust\ ςε 3( )t — sluçajnaq stupençataq funkcyq s uzlamy k l i i k ε ε βε ε, � = ∑   1 , = = ∑ 0 1 0 i , k = 0, … , n. Yzvestno [21], çto v sluçae ravnomernoho syl\noho peremeßyvanyq spravedlyva ocenka P β β ϕ ϕε ε ε εk k h h− ≥{ } ≤� 6 6( ) ( ) , k = 1, 2, … , n. Yspol\zuq πtu ocenku y ocenku (28), ubeΩdaemsq v tom, çto v¥polnqetsq ne- ravenstvo P sup ( ) ( ) 0 3 2 12 1 ≤ ≤ −− ≥ +   t T ht t ce T l ς ς ε εε ε γ ε ε        ≤ ≤ P max ( ) 1 1 11 12 1 ≤ ≤ + == −− ≥ +∑∑ k n i i i k i k hce nβ βε ε γ ε�      ≤ ≤ P β βε ε γ ε i i i n hce n− ≥ +      = −∑ � 1 12 1( ) ≤ ≤ P β β ε ε ε γ ε ε i i h i n ce T l ce− ≥{ } ≤ +    − = −∑ � 12 1 12 1 γγ εh . (29) Postroym analoh lomanoj ςε 3( )t , no uΩe s uzlamy, postroenn¥my po summam nezavysym¥x normal\no raspredelenn¥x velyçyn. Pust\ � �β βε ε 1 , ,…{ }n — vektor, sostoqwyj yz nezavysym¥x sluçajn¥x vely- çyn, a vektor φ φε ε 1 , ,…{ }n sostoyt yz nezavysym¥x normal\no raspredelenn¥x sluçajn¥x velyçyn, ymegwyx, kak y velyçyn¥ �βε 1 , … , �βε n , nulevoe srednee y ravn¥e vtor¥e moment¥, t.Fe. M Mφ βε ε i i= ˜ 0 , M M Mφ β βε ε ε i i i[ ] = [ ] = [ ]2 2 2˜ , i = 1, … , n. V postroenyqx A.FY.FSaxanenko [21 – 23] po φε n = φ φε ε 1 , ,…{ }n — vektoru yz ne- zavysym¥x normal\no raspredelenn¥x velyçyn — stroylsq ��βε n = �� β βε ε 1 , ,…{ }n ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 746 B. V. BONDAREV, S. M. KOZÁR\| — vektor yz nezavysym¥x velyçyn, ymegwyx to Ωe samoe fyksyrovannoe ras- predelenye, çto y ysxodn¥j vektor � �β βε ε 1 , ,…{ }n , no naybolee blyzkyj k vekto- ru φε n . Ymeet mesto y obratnoe postroenye (sm. [24]), kohda po ysxodnomu vek- toru � �β βε ε 1 , ,…{ }n stroytsq vektor φε n = φ φε ε 1 , ,…{ }n . Zametym, çto s veroqt- nost\g 1 βε k ≤ 2 Kbε (tohda y �βε k ≤ 2 Kbε s veroqtnost\g 1), a v sylu sledstvyq 1 yz [23] uslovye α β α β βε ε εM D˜ exp ˜ ˜ k k k 3 { } ≤ , k = 1, … , n, v¥polnqetsq s α = ( )4 1Kbε − . Yz rezul\tatov rabot¥ [24] (lemma 1.5) sleduet, çto moΩno zadat\ sluçaj- n¥e velyçyn¥ �βε 1 , … , �βε n , voobwe hovorq, na bolee bohatom veroqtnostnom prostranstve ′ ′ℑ ′( )Ω , , P y po πtoj posledovatel\nosty na veroqtnostnom pros- transtve ′ ′ℑ ′( )Ω , , P postroyt\ posledovatel\nost\ φε 1 , … , φε n nezavysym¥x normal\no raspredelenn¥x sluçajn¥x velyçyn takyx, çto ′ = =M M�β φε ε i i 0 y ′ =D D�β φε ε i i dlq vsex i, pryçem budet v¥polnqt\sq neravenstvo ′ { }M exp αc n∆ ≤ 1 + αBn , ∆n = max 1 1 1≤ ≤ = = ∑ ∑− k n i i k i i k �β φε , (30) hde c > 0 — nekotoraq absolgtnaq postoqnnaq, Bn 2 = ′=∑ D β̃ε kk n 1 ≤ 2 2 2nK bε . Ne umen\ßaq obwnosty (sm. [23]) moΩno sçytat\, çto 0 < c ≤ 1 64 . V teoreme Bekeßa – Fylyppa trebuetsq, çtob¥ prostranstvo, na kotorom za- dagtsq sluçajn¥e velyçyn¥, b¥lo dostatoçno bohat¥m; v protyvnom sluçae rassuΩdenyq provodqtsq na bolee bohatom prostranstve. ∏to obæqsnqetsq tem, çto dal\nejßye postroenyq provodqtsq çerez ravnomerno raspredelennug slu- çajnug velyçynu, kotorug nel\zq, naprymer, zadat\ na prostranstve s atomamy Ω = ω1{ , ω2 , … , ωn} (to Ωe samoe kasaetsq postroenn¥x v dal\nejßem nor- mal\no raspredelenn¥x velyçyn). M¥ yznaçal\no budem predpolahat\, çto ysxodnoe prostranstvo dostatoçno bohatoe, tohda vse postroenyq moΩno v¥polnyt\ na ysxodnom prostranstve. Pust\ ςε 4 ( )t — sluçajnaq stupençataq funkcyq s uzlamy k l i i k ε ε φε , = ∑   1 , = = ∑ 0 1 0 i , k = 0, … , n. Tohda, yspol\zuq neravenstvo (30), ymeem ocenku P sup ( ) ( ) 0 4 3 3 ≤ ≤ − >{ } t T t t xς ςε ε ≤ 1 64 641 1 1 3max ≤ ≤ = = − >       ∑ ∑ k n i i k i i k x�β φ ε ε ≤ ≤ exp exp−             x K b K b n 3 256 1 256ε εε ε M ∆ ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 747 ≤ exp / −       + +       x K b T l 3 1 2 256 1 1 2 2 1 ε εε ε   . V sylu toho, çto βε k = −νε k + ε ξ ξε ε εU k l h U kl(( ) ( )− +( ) − ( )[ ]1 , poluçaem M Mβ νε ε k k[ ] − [ ]2 2 ≤ M ν ε γ ε k Kc ( )2 1 2 4/ + ε γ 4 2 2 2 K c ≤ ≤ ε γε ψb D K Kc4 + ε γ 4 2 2 2 K c = ε εb D1 , (31) hde D1 = 16 2 2C cK K λγ + 4 2 2 2 K c γ , a poskol\ku M ν ε σε εk b  = 2 2 , k = 1, … , n, (32) yz (31) y (32) ymeem ocenku Mεφ ε σεk b2 2− ≤ ε εb D2 , k = 1, … , n, (33) hde D2 = 16 4 2exp K cK K λ λγ { } + 4 2 2 2 K c γ . Yz (33) poluçaem M φ ε σε ε ε ε k k T l b[ ] −   = [ ] ∑ 2 2 2 1 / ≤ ε εε ε 2 2 2b D T l ≤ εTD2 2 . (34) Oboznaçyv �φk = φ φ k kM 2 , postroym sluçajnug stupençatug funkcyg ςε 5( )t s uzlamy k l b i i k ε ε σ φε ε, 2 1 � = ∑    , = = ∑ 0 1 0 i , k = 0, … , n. Yspol\zuq ocenku dlq maksymuma summ nezavysym¥x normal\no raspredelen- n¥x sluçajn¥x velyçyn [25, s. 67], naxodym P sup ( ) ( ) 0 4 5 4 ≤ ≤ − >{ } t T t t xς ςε ε ≤ 4 2 2 4 1 P M b xi i i n εφ ε σ φε−    >      = ∑ � ≤ ≤ 4 2 4 2 2 2 2 2 1 exp exp−{ } −  = ∑zx z b bi i n M M ε σ εφ ε σ ε ε       . (35) ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 748 B. V. BONDAREV, S. M. KOZÁR\| Yz (35) s uçetom ocenky (34) poluçaem P sup ( ) ( ) 0 4 5 4 ≤ ≤ − >{ } t T t t xς ςε ε ≤ 4 2 4 2 2 2 2exp − +       zx z TD bεσ . (36) Mynymyzyruq pravug çast\ (36) po z > 0, ymeem ocenku P sup ( ) ( ) 0 4 5 4 ≤ ≤ − >{ } t T t t xς ςε ε ≤ 4 2 4 2 2 2 2exp −       x b TD εσ . (37) Poskol\ku v sylu rezul\tatov rabot¥ [26] suwestvuet standartn¥j vynerov- skyj process W tε ( ) , t ≥ 0, takoj, çto dlq lgboj �φ ε σεi b 2 , i = 1, … , n, ymeet mesto predstavlenye ε σ φεb i 2 � = W i bε εε σ2( ) – W i bε εε σ−( )( )1 2 , i = 1, … , n, oboznaçaq çerez ηε 6( )t sluçajnug stupençatug funkcyg s uzlamy k l W k bε ε σε ε ε, 2( )( ) , k = 0, … , n, s uçetom upomqnutoho predstavlenyq y (37) ubeΩdaemsq v spravedlyvosty ocenky P sup ( ) ( ) 0 6 3 3 4 ≤ ≤ − > +{ } t T t t x xς ςε ε ≤ ≤ exp / −       + +       x K b T l 3 1 2 256 1 1 2 2 1 ε εε ε   + −       4 2 4 2 2 2 2exp x b TD εσ . (38) Pust\ teper\ ηε 7( )t — sluçajnaq stupençataq funkcyq s uzlamy k l W k lε ε σε ε ε, ⋅( )( )2 , k = 0, … , n. V sylu toho, çto dlq normal\noj N (0, 1) velyçyn¥ ξ v¥polnqetsq nera- venstvo P ξ ≥{ }x ≤ 2 2 2 exp −       x , yspol\zuq neravenstvo dlq summ nezavysym¥x symmetryçn¥x sluçajn¥x vely- çyn [25], poluçaem ocenku P sup ( ) ( ) 0 7 6 5 ≤ ≤ − >{ } t T t t xς ςε ε ≤ P max ( ) ( ) 1 2 2 5≤ ≤ − >{ }k n W k b W k l xε ε ε εε σ ε σ ≤ ≤ P max ( ) 1 2 5≤ ≤ >{ }k n W k h xε εε σ ≤ 4 2 5P W n h xε εε σ( ) >{ } ≤ ≤ 4 2 5P W Th l xε ε ε σ    >      = 4 1 5P W x l Th ε ε ε ( ) >       ≤ ≤ 8 2 5 2 2exp − ( )         x TKD h lψ ε ε/ . (39) ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 749 Pust\ W tε σ( )2 = σ ε�W t( ) , hde �W tε ( ) = W tε σ σ( )/2 — standartn¥j vynerovskyj process. Tohda ymeet mesto ocenka P sup ( ) ( ) 0 7 6 ≤ ≤ − >{ } t T W t t xσ ςε ε� ≤ ≤ P max sup ( ) ( ) ( )1 1 2 1 ≤ ≤ − ≤ ≤ − − k n k l t k l W t W k ε ε ε ε ε ε σ εll xεσ2 6( ) >       + + P sup ( ) ( ) n l t T W t W n l x ε ε ε ε ε σ ε σ ≤ ≤ − >       2 2 6 ≤ ≤ P sup ( ) 0 2 6 1 ≤ ≤= >       ∑ τ ε ε ε τσ lk n W x + P sup ( ) [ / ]0 2 6 ≤ ≤ − >      τ ε ε ε ε τσ T nl W x ≤ ≤ n W x l P sup ( ) 0 2 6 ≤ ≤ >      τ ε ε ε τσ + P sup ( ) 0 2 6 ≤ ≤ >      τ ε ε ε τσ l W x ≤ ≤ T l W x lε τσ ε τ ε ε ε +    >      ≤ ≤ 1 0 2 6P sup ( ) ≤ ≤ T l W x lε τ σ ε τ ε ε ε +    >      ≤ ≤ 1 0 6 2P sup ( )� / ≤ ≤ 4 1 6 2T l W l x ε ε σ ε ε ε+    >{ }P � ( ) / ≤ ≤ 4 1 1 6 2T l P W x l ε σ ε ε ε ε+    >{ }� ( ) / ≤ 8 1 2 6 2 2 T l x KD lε εε ψ ε +    −         exp . (40) Obæedynqq ocenky (27), (29), (38) – (40), poluçaem ocenku P sup ( ) ( ) 0 6 5 3 4 ≤ ≤ − > + + +    t T W t t x x x xσ ςε ε� + + T l ce x Kch ε ε γε γ ε+    + +−1 12 2 4 1 ≤ 8 1 2 6 2 2 T l x KD lε εε ψ ε +    −         exp + + 8 2 5 2 2exp − ( )         x TKD h lψ ε ε/ + T l ce h ε ε γ ε+    −1 12 + + 4 2 4 2 2 2 2exp −       x b TD εσ + 4 1 2 1 2 2 T l x l KDε εε ε ψ +    −         exp + + exp / −       + +       x K b T l 3 1 2 256 1 1 2 2 1 ε εε ε   . V¥byraq xi = εδ , 0 < δ < 1 4/ , i = 1, … , 5, lε = 1/ ε , hε = 1 1 4/ε δ/ − , ymeem P sup ˜ ( ) ( ) / / 0 6 1 12 41 4 ≤ ≤ −− > + +     +       − t T W t t T ce Kcσ ς ε ε ε ε γε ε δ γ ε δ ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 750 B. V. BONDAREV, S. M. KOZÁR\| ≤ 8 1 1 2 2 1 2 2 T KDε εψ δ+     −         −exp / + 8 1 2 2 1 4 exp / −         −TKDψ δε + + T l ce ε ε γ ε δ +    − − 1 12 1 4/ / + 4 2 2 1 2 2 2 2 exp / −        − σ ε δ TD + + 4 1 1 21 2 2 2 T KDε ε δ ψ +     −         −exp / + + exp / / −       + +          − 1 256 1 1 2 2 1 1 2 1 2 K T ε εδ . Yz posledneho s uçetom (21) ymeem (20). TeoremaF2 dokazana. Sledstvye 1. V uslovyqx teorem¥ 2 spravedlyv¥ ocenky P sup ( ) ( ) ( ) ( ) / 0 0≤ ≤ ( ) ( ) + >∫ t T t s s dW s W tε ψ ξ β ξ σ ε ε� δδ ε γε +         4 cK ≤ γ ε . Sledstvye 2. Spravedlyva ocenka M sup ( ) ( ) 0 2 ≤ ≤ + t T m t W tς σε ε� ≤ max , ( , )δ γε ε 2 0 m D m T( ) , (41) hde D m T0( , ) = 1 + 4 4 4 1 2 12 2 m m m m mD K m m T mψ / ( )!! −     − + + 16 2 4 4 1 2 4 1 2 2 2 m m m m cK T m m m m γ     + −     −   ( )            m m mD Kψ 2 . Dejstvytel\no, tak kak yz ocenky (20) sleduet ocenka M sup ( ) ( ) 0 2 ≤ ≤ + t T m t W tς σε ε� ≤ δε 2m + γ ς σε ε εM sup ( ) ( ) / 0 4 1 2 ≤ ≤ +   t T m t W t� ≤ ≤ δε 2m + 4 0 4 2 0 m t T m m t T t Wγ ς σε ε εM Msup ( ) sup ( ≤ ≤ ≤ ≤     + � tt m ) / 4 1 2    ≤ ≤ δε 2m + 4 4 4 10 4 1 2 m t T m mt m m γ ς σε εM sup ( ) / ≤ ≤     + −     −         2 2 1 m mT m( )!! (42) y v sylu toho, çto v dannom sluçae spravedlyva ocenka ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 751 M sup ( ) ( ) ( ) / 0 0 4 ≤ ≤ ( ) ( )∫ τ τ ε ε ψ ξ β ξ T m s s dW s ≤ ≤ T m m m m D Km m m m m2 4 2 4 24 4 1 2 4 1 −     −      ( ) ψ (43) (v spravedlyvosty (43) netrudno ubedyt\sq, vospol\zovavßys\ ocenkoj dlq mo- mentov supremuma neprer¥vnoho martynhala [19, s. 174]), yspol\zuq sootno- ßenye sup ( ) 0≤ ≤t T tςε ≤ ε γ 2 cK + sup ( ) 0≤ ≤t T tηε , s uçetom (43) ymeem sup ( ) 0 4 ≤ ≤t T mtςε ≤ ε γ 2 4 1 4 2 2m m m cK−     + 24 1 0 4m t T mt− ≤ ≤ sup ( )ηε ≤ ≤ ε γ 2 4 1 4 2 2m m m cK−     + 2 4 4 1 2 4 14 1 2 4 2 4m m m m mT m m m m D K− −     −      ( ) ψ 22m . (44) Yz (42) s uçetom (44) poluçaem (41). Sledstvye 3. Spravedlyva ocenka M sup ( ) ( ) 0 2 ≤ ≤ + t T m t W tη σε ε� ≤ δ ε γε +    6 2 cK m + + γ η σε ε εM sup ( ) ( ) / 0 4 1 2 ≤ ≤ +   t T m t W t� ≤ δ ε γε +    6 2 cK m + + 4 0 4 4 0 4m t T m m t T m t W tγ η σε ε εM Msup ( ) sup ( ) ≤ ≤ ≤ ≤ +  �    1 2/ ≤ ≤ δ ε γε +    6 2 cK m + +F 4 4 4 1 2 4 12 2 2 4 2m m m m mT m m m m D Kγ ε ψ−     −      ( ) mm     + +F σ4 4 2 1 2 4 4 1 4 1m m mm m T m −     −     ( )!! / ≤ δ ε γε +    4 2 cK m + +F 4 4 4 1 2 2 1 4 1m m m mT m m m m m −     −( ) + − γ ε ( ) ( )!! DD Km m ψ 2 ≤ ≤ max , ( , , )δ ε γ γε ε+          4 2 1 cK D m T K m , hde ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 752 B. V. BONDAREV, S. M. KOZÁR\| D m T K1( , , ) = 1 + + 4 4 4 1 2 2 1 4 1m m m mT m m m m m D −     −( ) + − ( ) ( )!! ψ 22m mK . V zaklgçenye pryvedem nekotor¥e prymer¥. Prymer 1. Pust\ α( )x = sin2 1 2 πx − , β( )x = 1, f x( ) = cos2 πx , tohda ϑ π π( ) exp sinx x= { }1 2 2 , ϑ( )1 1= , G x0( ) = exp sin exp sin−           ∫ 1 2 2 1 2 2 0 1 π π π πx x dx , ρ( )x = exp sin exp sin−      −     ∫ 1 2 2 1 2 2 0 1 π π π πx x dx         −1 , f = 1 + 1 1 2 21 2 0 1 −( ) −              − − ∫e x dx/ exp sinπ π π 11 . Prymer 2. Pust\ α( )x = −1 , β( )x = 1, f x( ) = 2 2cos πx , tohda ϑ( )x = = e x2 , ϑ( )1 2= e , G x e e 0 2 2 1 1 ( ) = − + , ρ( )x ≡ 1 , f = 1. Prymer 3. Pust\ α( )x = 0 , β( )x = 1, f x( ) = 2 2cos πx , tohda ϑ( )x ≡ 1 , G x0( ) ≡ 1, ρ( )x ≡ 1 , f = 1. 1. Lypcer R. Í., Íyrqev A. N. Martynhal¥ y predel\n¥e teorem¥ dlq sluçajn¥x processov.// Ytohy nauky y texnyky. Sovr. probl. matematyky. Fundam. napravlenyq. – 1989. – 45. – S.F159 – 251. 2. Bondarev B. V. O neravenstve Kolmohorova – Haeka – Ren\y dlq normyrovann¥x yntehralov ot processov so slaboj zavysymost\g // Teoryq veroqtnostej y ee prymenenyq. – 1997. – V¥p. 2. – S. 225 – 238. 3. Bondar[v B. V., Zubko M. L. Pro ocinku ßvydkosti zbiΩnosti v pryncypi invariantnosti // Visn. Kyiv. un-tu. Fiz-mat. nauky. – 2001. – Vyp.F4. – S. 104 – 113. 4. Utev S. A. Neravenstva dlq summ slabozavysym¥x sluçajn¥x velyçyn y ocenky skorosty sxodymosty v pryncype ynvaryantnosty //Trud¥ Yn-ta matematyky SO AN SSSR. – 1984. – 3. – S. 50 – 77. 5. Dav¥dov G. A. O sxodymosty raspredelenyj, poroΩdenn¥x stacyonarn¥my sluçajn¥my processamy // Teoryq veroqtnostej y ee prymenenyq. – 1968. – 13, # 4. – S. 730 – 737. 6. Çykyn D. O. Funkcyonal\naq predel\naq teorema dlq stacyonarn¥x processov: martyn- hal\n¥j podxod // Tam Ωe. – 1989. – 14, # 4. – S. 731 – 741. 7. Ûakod Û., Íyrqev A. N. Predel\n¥e teorem¥ dlq sluçajn¥x processov.: Per. s anhl. – M.: Fyzmatlyt, 1994. – T. 2. – 368 s. 8. Skoroxod A. V. Asymptotyçeskye metod¥ teoryy stoxastyçeskyx dyfferencyal\n¥x urav- nenyj. – Kyev: Nauk. dumka, 1987. – 328 s. 9. Bondarev Boris V., Zoobko Maxim L. The application of the invariance principle for stationary se- quences with mixing // Prykl. statystyka. Aktuarna ta finansova matematyka. – 2001. – # 1. – S. 49 – 59. 10. Bondarev B. V., Koz¥r\ S. M. Ob ocenke skorosty sblyΩenyq reßenyq ob¥knovennoho dyf- ferencyal\noho uravnenyq, vozmuwennoho fyzyçeskym bel¥m ßumom, y reßenyq soot- vetstvugweho uravnenyq Yto. I // Tam Ωe. – 2006. – # 2. – S. 63 – 91. ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 PEREMEÍYVANYE „PO YBRAHYMOVU”. OCENKA SKOROSTY SBLYÛENYQ … 753 11. Bondarev B. V., Koz¥r\ S. M. Ob ocenke skorosty sblyΩenyq reßenyq ob¥knovennoho dyf- ferencyal\noho uravnenyq, vozmuwennoho fyzyçeskym bel¥m ßumom, y reßenyq soot- vetstvugweho uravnenyq Yto. 2 // Tam Ωe. – 2007. – #1. – S. 68 – 97. 12. Bondarev B. V., Kovtun E. E. Ocenka skorosty sblyΩenyq v ob¥knovenn¥x dyfferen- cyal\n¥x uravnenyqx, naxodqwyxsq pod vozdejstvyem sluçajn¥x processov s b¥str¥m vremenem // Ukr. mat Ωurn. – 2005. – 57, # 4. – S. 435 – 457. 13. Bondarev B. V., Kovtun E. E. Pryncyp ynvaryantnosty dlq stacyonarn¥x processov. Ocenka skorosty sxodymosty // Visn. Donec. un-tu. Ser. A. – 2004. – # 1. – S. 31 – 55. 14. Safonova O. A. Ob asymptotyçeskom povedenyy yntehral\n¥x funkcyonalov ot dyffu- zyonn¥x processov s peryodyçeskymy koπffycyentamy // Ukr. mat. Ωurn. – 1992. – 44, # 2. – S. 245 – 252. 15. Ûurbenko Y. H. Analyz stacyonarn¥x y odnorodn¥x sluçajn¥x system. – M.: Yzd-vo Mosk. un-ta, 1987. – 240 s. 16. Bensoussan A., Lions J.-L., Papanicolau G. Asymptotic analysis for periodic structures. – North- Holland Publ. Comp., 1978. – 700 p. 17. Ybrahymov Y. A., Lynnyk G. V. Nezavysym¥e y stacyonarno-svqzann¥e velyçyn¥. – M.: Nauka, 1965. – 524 s. 18. Hyxman Y. Y., Skoroxod A. V. Stoxastyçeskye dyfferencyal\n¥e uravnenyq. – Kyev: Nauk. dumka, 1968. – 354 s. 19. Hyxman Y. Y., Skoroxod A. V. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y yx prylo- Ωenyq. – Kyev: Nauk. dumka, 1982. – 612 s. 20. Berkes I., Philipp W. Approximation theorem for independent and weakdependent random vektors // Ann. Probab. – 1979. – 1, # 1. – P. 29 – 54. 21. Saxanenko A. Y. Skorost\ sxodymosty v pryncype ynvaryantnosty dlq raznoraspredelen- n¥x velyçyn s πksponencyal\n¥my momentamy // Trud¥ Yn-ta matematyky SO AN SSSR. – 1984. – 3. – S. 4 – 49. 22. Saxanenko A. Y. Ocenky v pryncype ynvaryantnosty // Tam Ωe. – 1985. – 5. – S. 27 – 44. 23. Saxanenko A. Y. O toçnosty normal\noj approksymacyy v pryncype ynvaryantnosty // Tam Ωe. – 1989. – 13. – S. 40 – 66. 24. Bondarev B. V., Kolosov A. A. K voprosu o pryncype ynvaryantnosty dlq slabozavysym¥x sluçajn¥x velyçyn // Prykl. statystyka. Aktuarna ta finansova matematyka. – 2002. – # 2. – S. 63 – 71. 25. Petrov V. V. Summ¥ nezavysym¥x sluçajn¥x velyçyn. – M.: Nauka, 1972. – 414 s. 26. Kolosov A. A. O postroenyy vynerovskoho processa po koneçnoj posledovatel\nosty neza- vysym¥x haussovskyx velyçyn // Prykladna statystyka. Aktuarna ta finansova matematyka. – 2007. – # 1. – S. 97 – 101. Poluçeno 21.10.08, posle dorabotky — 01.04.10 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
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institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
last_indexed	2026-03-24T02:32:33Z
publishDate	2010
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-29052020-03-18T19:40:12Z Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I Перемешивание „по Ибрагимову". Оценка скорости сближения семейства интегральных функционалов от решения дифференциального уравнения с периодическими коэффициентами с семейством вииеровских процессов. Некоторые приложения. I Bondarev, B. V. Kozyr', S. M. Бондарев, Б. В. Козырь, С. М. Бондарев, Б. В. Козырь, С. М. We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space $C[0, T]$. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation. Доведено, що обмежена 1-періодична функція від розв'язку однорідного за часом дифузійного рівняння з 1-періодичними коефіцієнтами утворює процес, що задовольняє умову рівномірного сильного перемішування. Встановлено оцінку швидкості зближення за ймовірністю в метриці простору $C[0, T]$ деякого нормованого інтегрального функціонала від розв'язку звичайного однорідного за часом стохастичного диференціального рівняння з 1-періодичними коефіцієнтами з сім'єю віперових процесів. Як приклад, розглянуто звичайне диференціальне рівняння, збурене швидкоосцилюючим центрованим процесом, який є 1-періодичною функцією від розв'язку однорідного за часом стохастичного диференціального рівняння з 1-періодичними коефіцієнтами. Встановлено оцінку швидкості зближення розв'язку такого рівняння з розв'язком відповідного стохастичного рівняння Іто. Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2905 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 733–753 Український математичний журнал; Том 62 № 6 (2010); 733–753 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2905/2561 https://umj.imath.kiev.ua/index.php/umj/article/view/2905/2562 Copyright (c) 2010 Bondarev B. V.; Kozyr' S. M.
spellingShingle	Bondarev, B. V. Kozyr', S. M. Бондарев, Б. В. Козырь, С. М. Бондарев, Б. В. Козырь, С. М. Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title	Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title_alt	Перемешивание „по Ибрагимову". Оценка скорости сближения семейства интегральных функционалов от решения дифференциального уравнения с периодическими коэффициентами с семейством вииеровских процессов. Некоторые приложения. I
title_full	Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title_fullStr	Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title_full_unstemmed	Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title_short	Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
title_sort	mixing “in the sense of ibragimov.” estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. some applications. i
url	https://umj.imath.kiev.ua/index.php/umj/article/view/2905
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Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

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