Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell, Nadine Guillotin-Plantard, and Françoise Pène

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Abstract

Random walks in random scenery are processes defined by $Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\cdots+X_{k}}$, where $(X_{k},k\ge1)$ and $(\xi_{y},y\in{\mathbb{Z}}^{d})$ are two independent sequences of i.i.d. random variables with values in ${\mathbb{Z}}^{d}$ and $\mathbb{R}$ respectively. We suppose that the distributions of $X_{1}$ and $\xi_{0}$ belong to the normal basin of attraction of stable distribution of index $\alpha\in(0,2]$ and $\beta\in(0,2]$. When $d=1$ and $\alpha\ne1$, a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete 50 (1979) 5–25) and a local limit theorem in (Ann. Probab. To appear). In this paper, we establish the convergence in distribution and a local limit theorem when $\alpha=d$ (i.e. $\alpha=d=1$ or $\alpha=d=2$) and $\beta\in(0,2]$. Let us mention that functional limit theorems have been established in (Ann. Probab. 17 (1989) 108–115) and recently in (An asymptotic variance of the self-intersections of random walks. Preprint) in the particular case when $\beta=2$ (respectively for $\alpha=d=2$ and $\alpha=d=1$).

Résumé

Les promenades aléatoires en paysage aléatoire sont des processus définis par $Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\cdots+X_{k}}$, où $(X_{k},k\geq1)$ et $(\xi_{y},y\in\mathbb{Z}^{d})$ sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans $\mathbb{Z}^{d}$ et $\mathbb{R}$ respectivement. Nous supposons que les lois de $X_{1}$ et $\xi_{0}$ appartiennent au domaine d’attraction normal de lois stables d’indice $\alpha\in(0,2]$ et $\beta\in(0,2]$. Quand $d=1$ et $\alpha\neq1$, un théorème limite fonctionnel a été prouvé dans (Z. Wahrsch. Verw. Gebiete 50 (1979) 5–25) et un théorème limite local dans (Ann. Probab. To appear). Dans ce papier, nous prouvons la convergence en loi et un théorème limite local quand $\alpha=d$ (i.e. $\alpha=d=1$ ou $\alpha=d=2$) et $\beta\in(0,2]$. Mentionnons que des théorèmes limites fonctionnels ont été établis dans (Ann. Probab. 17 (1989) 108–115) et récemment dans (An asymptotic variance of the self-intersections of random walks. Preprint) dans le cas particulier où $\beta=2$ (respectivement pour $\alpha=d=2$ et $\alpha=d=1$).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 2 (2013), 506-528.

Dates
First available in Project Euclid: 16 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1366117656

Digital Object Identifier
doi:10.1214/11-AIHP466

Mathematical Reviews number (MathSciNet)
MR3088379

Zentralblatt MATH identifier
1278.60046

Subjects
Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes

Keywords
Random walk in random scenery Local limit theorem Local time Stable process

Citation

Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise. Limit theorems for one and two-dimensional random walks in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 2, 506--528. doi:10.1214/11-AIHP466. https://projecteuclid.org/euclid.aihp/1366117656


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