Limit theorems for one and two-dimensional random walks in random scenery
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, p. 506-528
Les promenades aléatoires en paysage aléatoire sont des processus définis par ${Z}_{n}:={\sum }_{k=1}^{n}{\xi }_{{X}_{1}+\cdots +{X}_{k}}$, où $\left({X}_{k},k\ge 1\right)$ et $\left({\xi }_{y},y\in {ℤ}^{d}\right)$ sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans ${ℤ}^{d}$ et $ℝ$ respectivement. Nous supposons que les lois de ${X}_{1}$ et ${\xi }_{0}$ appartiennent au domaine d’attraction normal de lois stables d’indice $\alpha \in \left(0,2\right]$ et $\beta \in \left(0,2\right]$. Quand $d=1$ et $\alpha \ne 1$, 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 $\beta \in \left(0,2\right]$. 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 $\alpha =d=1$).
Random walks in random scenery are processes defined by ${Z}_{n}:={\sum }_{k=1}^{n}{\xi }_{{X}_{1}+\cdots +{X}_{k}}$, where $\left({X}_{k},k\ge 1\right)$ and $\left({\xi }_{y},y\in {ℤ}^{d}\right)$ are two independent sequences of i.i.d. random variables with values in ${ℤ}^{d}$ and $ℝ$ respectively. We suppose that the distributions of ${X}_{1}$ and ${\xi }_{0}$ belong to the normal basin of attraction of stable distribution of index $\alpha \in \left(0,2\right]$ and $\beta \in \left(0,2\right]$. When $d=1$ and $\alpha \ne 1$, 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 $\beta \in \left(0,2\right]$. 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 $\alpha =d=1$).
DOI : https://doi.org/10.1214/11-AIHP466
Classification:  60F05,  60G52
@article{AIHPB_2013__49_2_506_0,
author = {Castell, Fabienne and Guillotin-Plantard, Nadine and P\ene, Fran\c coise},
title = {Limit theorems for one and two-dimensional random walks in random scenery},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {49},
number = {2},
year = {2013},
pages = {506-528},
doi = {10.1214/11-AIHP466},
zbl = {1278.60046},
mrnumber = {3088379},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2013__49_2_506_0}
}

Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise. Limit theorems for one and two-dimensional random walks in random scenery. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 506-528. doi : 10.1214/11-AIHP466. http://www.numdam.org/item/AIHPB_2013__49_2_506_0/`

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