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

Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 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 $(X_k,k\ge 1)$ 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 $ℝ$ 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 $\alpha \ne 1$, 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 $\beta \in (0,2]$. 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 $\alpha =d=1$).

Random walks in random scenery are processes defined by $Z_n:={\sum }_{k=1}^n{\xi }_{X_1+\cdots +X_k}$, where $(X_k,k\ge 1)$ 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 $ℝ$ 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 $\alpha \ne 1$, 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 $\beta \in (0,2]$. 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 $\alpha =d=1$).

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