Limit theorems for stochastic processes of jump type as currentvalued processes

(1)

Author(s) MATSUMOTO, Hiroyuki

Citation [岐阜大学教養部研究報告] vol.[22] p.[101]-[105]

Issue Date 1986

Rights

Version 岐阜大学教養部 (Dept. of Math. Fac. of Gen. Educ., Gifu Univ.)

URL http://hdl.handle.net/20.500.12099/47596

※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。

(2)

Dept. of M ath. F ac. of Gen. E duc. , Gifu U niv ( Received Oct. 13, 1986)

1. 1ntroduction

Recently sQveral authors have studied about infinite dimensional stochastic processes, in particular, about current valued stochastic processes (cf. [ L] , [21, [51, [6] and references

therein). ln thisnoteweconsider random walkswith continuous timeφarameter on a

torus defined as solutions of stochastic differential equations of j um p type (see ( 1) in the next sed ion) and study lim it theorem s for the current valued processes defined through the line integrals of smooth differentia1 1-forms along the jumps of the random walks。

T he p.urpose of this note is to show a sort of hom oginization (T heorem 1) and the invariance principle (T heorem 2) for our current valued stochastic processesレ

2. R esults

H iroyuki M A T SU M OT O

101

L imit theorems for stochastic processes of j ump type as current valued processes

Dedicated to Professor K . 0 hta on his 60th birthday

ln this note w e consider stochastic processes { λ? } a 0, 7zE N , of jum p type defined as

f0110ws. Let { ら ,…,亀} bethestandardba51sof R“, び= { 土1,…, 土の, ら = -にxxfor zxく 0

and ジ be the uniform probability measure on a W e consider a Poisson point process

畑 with a charad eristic measure が 1ノ( 面 ) defined on a probability space (亀 J √F ) for

each 77E N (see, lkeda- W atanabe [3]), L et us consider the following stochasticdifferential equation of jump type :

λ? = XE Rべ

where A臨 is the counting measure of the point function 九 . T hen it is easy to see that

(1) has a uniquesolution X”= { X7雇 o(cf. 圖 ) andXり convergesinlaw toastandard Brown-

ian m otion on 7? ゛ as 刄 tends to cx) 。

Let の 1be the space of all smooth differentia1 1-forms on R “ with periodic qoeffidents,

i.e., の1= { α= Σ穴1αf(x)ぱが; φ(x)∈(グ(R“) andα心十応)= 哨(ズ) foreveryxE R゛ andれノ= 1,

…,の. From now onweregardの1andλ? asthespaceof all smoothdifferentia11-forms

aud a stoc!lastic process on a j - dimensional torus T(1, resped ively, in the natural way.

W e endow £ ) 1 with the Schw arz topology.

聊=刀十ズヱ te uN陥(面,ゐ) (1)

(3)

九,い)a =fjJ naX u x(j十゛″ )力

where

and

い

y

Nh (加 , 冶)= 越衣加 , 冶) - が lノ(加 ) 血

T hen it is easy to see that M 71= { 訂 7雇 ois a の 1- valued stochastic process, where の 1=

の1( F ) is thedual space of のl with resped to the Schwarz topology on 度)1. M oreover

let us define, for λ> 0, な心 = い肝侑 aoby な y = λ- 1/2訂外 and denote by Q夕 the probability law of 刄 o on f ) ([0, (x⊃) 濯 ) i) . H ere p ([0, 00] ; 度) 1) isthespace of a11の 1-valued paths

on [O戸 ] which areright-continuouswith left handlimitswithresped tothetopology of の1, and the topology of this 5pace is the Skorohod topology.

W e are interested in the limit theorems for 研 as λ and n tend to 心 F irstly let us

-

fix x2 and study the case that λ tends to 皿 D enote by Q″the probability law on C ( ( 0,

∽) 汐) 1), the space of a11ヱ)1-valued continuouspathson [0,(x)), of a W iener の1-process

whose mean functional is zero and covariance functional in the sense of lt6 [1] isgiven by N ow define, for α= Σ 匹 1 αf(めみ ∈ £ ) 1, 肛 7( α) =

ヱ{‰。Z り2}岫越”(加ʼふ )ʼ

X ZXE び

/ Σ

for α= ΣL I αz x(ズ)冶 “ and β= ΣL 1βj x) 冶 “E j) 1, whereΣ denotesthesummationover

the state space of the random w alk χ 仇 VVe regard 研 a5 A probability measure on 力 ((0,

伺 ; 爪 ). ゛

N ow w e can state one of our results.

-

THEOREM 1. Q夕convergesweakly to Q″on 刀([0,∽] 沼) 1) asλtendsto (x⊃for each ふ

T he proof of T heorem l w ill be given in the next sed ion.

Secondly let us fix λ F or simplid ty of notations, w e consider here only the case λ=

1. B efore stating the result, w e remember the invariance principle, w hich says that, as 刀 tends to ∽ , the stochastic process χ z7= { χ 7Daodefined at the beginning of this section

convergesinlaw toastandardBrownianmotionχ= { χ函aoon Ty

L et us introduce the limiting stochastic process. L et Ql be the probability law of a

ヱ)1-valued continuous stochastic process whose mean functional is zero hnd covariance

functional in the sense of [1] is given by

for α,βE jど ) 1. Here £ denotes theexpection with resped to the probability law of the

standard Brownian motion χ and < ・, ・ > (x) is the natural i皿 er product in the cotangent space at xE F . N ote that the sample path j 7(α) = { 漏 (α)旨 oof the 1-dimensional mar- gi゛ l pl゛ocess of Ql is gi゛ ell by the stoch stic 1111e illteg9 1 310昭 the p21ths of X ʼ 工 lo。j ʼ For stochastic line integrals along the paths of diffusion processes, see lkeda- W atanabe

[31. Weregard Ql asaprobability measureon 詞 [O芦]) ; 払 )

2£[仁 <α ,β >(馬)力 ]

1/μ

て

βゆ | (X+ 7亀) 力)

Σ がー“ ( αU l(ズ十肥。) 力) (

(4)

p 目封だ(α)一封y (釧 4)≦尺1卜列2十和 | /-slλ -1み-2

for any s and j, where F denotes the eχped ation w ith resped to 戸 .

Proof. Because of the stationarity of { yWy (α)這 o, it is sufficient to show in case s= 0 T o avoid unnecessary troubles in notation, w e denote by 戸 (ズ,g; α) the integra卜

LEMMA F or any α∈ の I , there eχist positive constants 尺 l and 瓦 , independent of λ and 馬 such that

jiχ ,u ,n )

心

で

L皿 it theorems for stochastic processes of jump type as current valued processes 103

3. P roof of T heorem l

Firstly we prove the tightness of the family { Qタド o of probability measures on D ([0,

(x)) ; の1). By virtue of Mitomaʼscriterion, Theorem 4.10f [51, it issufficient to show that

the fam ily of probability m easures on 刀 ( (O芦 ) ; R 1) induced by the one- dimensional mar- ginal processes of Q夕 is tight. M oreover, by the standard probabilistic arguments (see, for example, the proof of Proposition 5.2 0f [4]) , this can be proved by the follow ing es- timate.

N ow we can state our second result.

T HEOREM 2. Qt converges w eakly to Ql on £) ((0, ∽ ) ぱ ) 1) as 77 tends to ∽ .

T he proof of T heorem 2 w ill be given in sed ion 4.

Finally w e give som e related remarks. Denote by 刄 = { y哨 } a othe sample path of Q1,

define, for λ> 0, j f = { y剛雇 oby y剛 = λ-1/りみ andlet Q be the probability law oL Mλ

on C((O匹 ) ; の1). Notethat G istheweak limit of { Qn 。 1,2,…. Moreover let Q bethe

probability law of a の 1- valued continuous stochastic process whose mean functional is zero and covariance functional is givenby 2( tZ¥Sj Td< α,β> (ぞ)心 ,where 寂 isthenormar- ized L ebegue measure on F . T hen Ochi [6] shows that G converges weakly to Q on C

([0,(x⊃ ) ; の1) asλtendsto 心

・ r -

M oreover w e can show that Q″, the limiting m easure in T heorem 1, converges weakly

to Q on C([0,(x)) 心) 1) asn tendsto (x). Consequently, for thefamily { Q牡 λ> Oandn=

1,2, … , of probability measures on 刀 ([0, ∽ ] ; の 1) , the tw o limiting operations, 筧→ X and λ→ ∽ , are commutative.

口″ (Xぶ;α)| ≦SUpl萌(X) | リ

Since w e have

---

we obtain, by using Doobʼs inequality

α =

α|副 (X十昭Z X)加

十 T yjT八田(X乱彫a)}2が沁か)庇

N ow w e have, by lt6ʼs formula,

いがʼ‰川 2= am7 図泌gde面匝meanzeyo

(5)

N eχt, by using lt6ʼs formula again, we have

l訂回(α)14= amayt鏑gde戒thmeauzeyo

十万耶(財回 (α )十分1 ʼ2μ (X7,Z j;α ))4-(訂?(α ))4

- 4(冊封(α))37“(X7,zx;α)れ1/2} が1ノ(面 )五

T hen, noting that the simple inequality

(α十,ろ)4- α4一如 3ゐ≦12α2ゐ2+ 12ゐ4

holds for any real とz and ろ, we get

戸 [|冊叩[釧 4]

p [ SUp l訂回(朔 2) ≦4p [ | 訂μ(α)12] ≦4SUplαf(X) μ (4)

£仁心I(Xg十Z 心)ゐ押2y (面)血

N ow note that χ 72= { χ 7} , aois a symmetric random w alk on a finite set and, therefore, that χ “ is ergodic and the invariant m easure is the uniform probability m easure. T hen

we see that < yぼ″以(が), Af″ʼ゛(y ) > xconverges almost surely to

j多忍{ズ1/‰に|(x十 z 心)面}{ズ1 /‰ん|(x十z ノら)ゐ} Th ep ro o fisc o m p le te d .

=球呪{jミフa‰I(Xg十 IX e u )ゐ}{

≦ P[12エハノ jシ λ -I(7“(Xいt鴇 ))2(なだ(α ))‰21 ノ (加)力 ] 十 P[1ヱD-2(7″ (χ 乳 n ;a))柚21 ノ (面)力].

Combined with (3) and (4) , this implies (2) in case s= O and, therefore, the assertion of the lemma.

N ow w e tum to the proof of T heorem 1.

Proof of T heorem 1. Since the tightness of { Q夕}5 0 1s proved by combining M itom aʼs result and the lemm a as mentioned at the beginning of this sed ion, w e have only to show the uniqueness of the limit law . F or this w e show that, for α1,… , が ∈ £ ) 1, 加 dimensional

̲ ̲

process J ʼ‰ 1,… y ) = { (訂回 (α1), づ肝゛(♂ ))‰ oconvergesin law to{ (培 (ぷ), … ,訂 7(が ))

- 一

良o,where{ 訂7に oisthesamplepathof Q″. Sincewehaveshownthetightnessof { Qr}

わo, the tightness of the se(4uence of the probability m easures induced by these ん- dim en- sional processes is clear. T herefore it is sufficie皿 to prove the convergence of finite dimensional distributions. T his can be proved if w e show that the quadratic variation process

{ < yぼ″μ(y),yぼo (y ) > ,;ぴ= 1,…,んhとoof j F ʼA(α1,…,ak) convergesinlaw tw to{ < yぼ″(y),

-

yぼ“{αう> j ,プ= 1,…,ん},ao・

ぺVe have

< yぼ“・λ (が), yぼ心(y ) > x

(6)

4. P roof of T heorem 2

Proof of T heorem 2. lf we put λ= 1 1n the lemma in the previous section, the tightness

o球 Q?} 。 1,2,… canbeestablished. Thereforewehaveonlytoshow theuniquenessof the

limiting measure, which follow s if w e show that the quadratic variation process j f″(α1,

…,α勺= { < yぼ”(が), A? 1(y ) > , ;y,ノ= 1,…,ん},aoconvergesinlaw to j f (α1,…, α゛) = { < yぼ(が), yぼ(y ) > , ; ぴ= 1,…,ノ覗aofor any α1,…,が∈£)1.

1Ve have 二

< 訂 ″ (が), 訂 “(y ) > ,

105 Limit theorems for stochastic processes of jump type as current valued processes

=で工{ズ1 /‰しI(X;j十四)d 削£ʼ‰柚(X;jナZ 心)み}が1 ノ(面)碍二 2£<ai,a j>(Xg )ゐ十〇 (征

[1] lto,K., Foundationsof stochasticdifferential equationsininfinitedimensional spaces, CBMS-NSF, RegionaI Conference Series in A pplied M athematics.

[2] lkeda,N. andOchi,Y., Central limittheoremsandrandom currents, LecturenotesinContnandlnform.

Sciences, 78 (1986)。

[3] lkeda,N. andχVatanabe,S., Stochasticdifferential equationsanddiffusionprocesses, North-Holland, A msterdam, 1981.

[4] Matsumoto,H. and Shigekawa,I., Limittheoremsfor stochasticflowsof diffeomorphismsofjumptype, Z. W ahr. 69 (1985), 507-540.

[5] Mitoma,I。 Tightnessof probabilitiesonC([0,1] ; y) ʼ) andD([0,1] ; が ), Ann. Probability 11 (1983), 989- 999.

[6] Ochi,Y., Limit theoremsfor aclassof diffusionprocesses, Stochastics, 15 (1985), 251-269.

R eferences

Therefore, since{ χ7} ,aoconvergesinlaw to{ 罵} ,ao, weseethatM71(αI,…,♂) converges

in law to j 7 (α1,… , α゛) . T he proof is completed.