LIMIT THEOREMS FOR ONE AND TWO DIMENSIONAL RANDOM WALKS IN RANDOM SCENERY

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LIMIT THEOREMS FOR ONE AND TWO-DIMENSIONAL RANDOM WALKS IN RANDOM SCENERY FABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, AND FRANÇOISE PÈNE Abstract. Random walks in random scenery are processes defined by Zn := ∑n k=1 ?X1+...+Xk , where (Xk, k ≥ 1) and (?y, y ? Zd) are two independent sequences of i.i.d. random variables with values in Zd and R respectively. We suppose that the distributions of X1 and ?0 belong to the normal basin of attraction of stable distribution of index ? ? (0, 2] and ? ? (0, 2]. When d = 1 and ? 6= 1, a functional limit theorem has been established in [16] and a local limit theorem in [7]. In this paper, we establish the convergence in distribution and a local limit theorem when ? = d (i.e. ? = d = 1 or ? = d = 2) and ? ? (0, 2]. Let us mention that functional limit theorems have been established in [3] and recently in [10] in the particular case when ? = 2 (respectively for ? = d = 2 and ? = d = 1). 1. Introduction Random walks in random scenery (RWRS) are simple models of processes in disordered media with long-range correlations.

• zn


• skorohod j1- topology


• when ?


• local limit


• random scenery


• characteristic function


• stable distribution


• limit theorems

Sujets

• Characteristic function
• Stable distribution
• Limit theorem

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 26
• Langue: English

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LIMIT THEOREMS FOR ONE AND TWO-DIMENSIONAL RANDOM WALKS IN RANDOM SCENERY

FABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, AND FRANÇOISE PÈNE

Abstract.Random walks in random scenery are processes defined byZ n := P kn =1 ξ X 1 +...+X k , where(X k , k≥1)and(ξ y , y∈Z d ) variables randomare two independent sequences of i.i.d. with values inZ d andRrespectively. We suppose that the distributions ofX 1 andξ 0 belong to the normal basin of attraction of stable distribution of indexα∈(0,2]andβ∈(0,2]. When d= 1andα6= 1, a functional limit theorem has been established in [16] and a local limit theorem in [7]. In this paper, we establish the convergence in distribution and a local limit theorem whenα=d(i.e.α=d= 1orα=d= 2) andβ∈(0,2]. Let us mention that functional limit theorems have been established in [3] and recently in [10] in the particular case whenβ= 2(respectively forα=d= 2andα=d= 1).

1.Introduction

Random walks in random scenery (RWRS) are simple models of processes in disordered media with long-range correlations. They have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [21], diffusion with random sources, or spin depolarization in random fields (we refer the reader to Le Doussal’s review paper [17] for a discussion of these models).

On the mathematical side, motivated by the construction of new self-similar processes with stationary increments, Kesten and Spitzer [16] and Borodin [4, 5] introduced RWRS in dimension one and proved functional limit theorems. This study has been completed in many works, in particular in [3] and [10]. These processes are defined as follows. Letξ:= (ξ y , y∈Z d )and X:= (X k , k≥1)be two independent sequences of independent identically distributed random variables taking values inRandZ d  sequencerespectively. Theξis called therandom scenery. The sequenceXis the sequence of increments of therandom walk(S n , n≥0)defined byS 0 := 0 andS n := P n i=1 X i , forn≥1. Therandom walk in random sceneryZis then defined by n−1 Z 0 := 0and∀n≥1, Z n := X ξ S k . k=0 Denoting byN n (y)the local time of the random walkS: N n (y) := #{k= 0, ..., n−1 :S k =y}, it is straightforward to see thatZ n can be rewritten asZ n = P y ξ y N n (y). As in [16], the distribution ofξ 0 is assumed to belong to the normal domain of attraction of a strictly stable distributionS β of indexβ∈(0,2], with characteristic functionφgiven by φ(u) =e −|u| β (A 1 +iA 2 sgn(u)) u∈R,

2000Mathematics Subject Classification.60F05; 60G52. Key words and phrases.Random walk in random scenery; local limit theorem; local time; stable process This research was supported by the french ANR project MEMEMO2. 1

LIMIT THEOREMS FOR 1-D AND 2-D RWRS 2 where0< A 1 <∞and|A 1 −1 A 2 | ≤ |tan(πβ/2)| will denote by. Weϕ ξ the characteristic function of theξ x ’s. Whenβ >1, this implies thatE[ξ 0 ] = 0. Whenβ= 1, we will further assume the symmetry condition sup  E  ξ 0 1I {|ξ 0 |≤t}   <+∞.(1) t>0 Under these conditions (forβ∈(0; 2]), there existsC ξ >0such that we have ∀t >0,P(|ξ 0 | ≥t)≤C ξ t −β .(2) Concerning the random walk, the distribution ofX 1 is assumed to belong to the normal basin of attraction of a stable distributionS 0 α with indexα∈(0,2]. Then the following weak convergences hold in the space of càdlàg real-valued functions defined on[0,∞)and onRrespectively, endowed with the SkorohodJ 1 -topology (see [2, chapter 3]) :  n −1/α S bntc  t≥0 n = → L ⇒ ∞ ( (t)) t≥0 U and   n − 1 β bn k X = x 0 c ξ ke 1   0 n = L → ⇒ ∞ (Y(x)) x≥0 ,withe 1 = (1,0,∙ ∙ ∙,0)∈Z d , x≥ whereUandYare two independent Lévy processes such thatU(0) = 0,Y(0) = 0,U(1)has distributionS 0 α ,Y(1)has distributionS β . Functional limit theorem. Our first result is concerned with a limit theorem for(Z [nt] ) t≥0  speaking,. Intuitively •whenα < d, the random walkS n is transient, its range is of ordern, andZ n has the same behaviour as a sum of aboutnindependent random variables with the same distribution as the variablesξ x . It was proved in [5] that forβ= 2,n −1/β (Z [nt] ) t≥0 converges in distribution in the spaceD([0,∞))of càdlàg functions endowed with the SkorohodJ 1 - topology, to a multiple of the process(Y t ). The caseβ∈(0,2]was also mentioned in [16] (see Remark 3). Whenβ <1is positive, a functional limit theorem inand the scenery the spaceD([0,∞))endowed with the SkorohodM 1 -topology, is proved in [1] or [14]. •whenα > d(i.e.d= 1and1< α≤2), the random walkS n is recurrent, its range is of ordern 1/α , its local times are of ordern 1−1/α , so thatZ n is of ordern 1− 1 α + α 1 β . In this situation, [16] and [4] proved a functional limit theorem forn −(1− α 1 + α 1 β ) (Z [nt] , t≥0) in the spaceC([0,∞))of continuous functions endowed with the uniform topology, the limiting process being a self-similar process, but not a stable one. •whenα=d(i.e.α=d= 1, orα=d= 2),S n is recurrent, its range is of order −1 n/log(n), its local times are of orderlog(n)so thatZ n is of ordern β 1 log(n) β β  this. In situation, a functional limit theorem in the space of continuous functions was proved in [3] ford=α=β= 2, and in [10] ford=α= 1andβ= 2.

Our first result gives a limit theorem forα=dand for any value ofβ∈(0; 2). We establish the convergence in the sense of finite distributions, and prove that the convergence in distribution does not hold for theJ 1 -topology whenβ6= 2but that the convergence in distribution holds for theM 1 -topology whenβ6= 1(for technical reasons, our proof does not apply whenβ= 1). Theorem 1.Letβ∈(0; 2) assume that the random walk is strongly aperiodic and that. We

(a)eitherd= 2andX 1 is centered, square integrable with invertible variance matrixΣand then we defineA:= 2 √ det Σ;