Limit theorems for one and two-dimensional random walks in random scenery
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 506-528.

Random walks in random scenery are processes defined by Z n := k=1 n ξ X 1 ++X k , where (X k ,k1) and (ξ y ,y d ) are two independent sequences of i.i.d. random variables with values in d and respectively. We suppose that the distributions of X 1 and ξ 0 belong to the normal basin of attraction of stable distribution of index α(0,2] and β(0,2]. When d=1 and α1, 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 α=d (i.e. α=d=1 or α=d=2) and β(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 β=2 (respectively for α=d=2 and α=d=1).

Les promenades aléatoires en paysage aléatoire sont des processus définis par Z n := k=1 n ξ X 1 ++X k , où (X k ,k1) et (ξ y ,y d ) sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans d et respectivement. Nous supposons que les lois de X 1 et ξ 0 appartiennent au domaine d’attraction normal de lois stables d’indice α(0,2] et β(0,2]. Quand d=1 et α1, 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 α=d (i.e. α=d=1 ou α=d=2) et β(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 β=2 (respectivement pour α=d=2 et α=d=1).

DOI: 10.1214/11-AIHP466
Classification: 60F05,  60G52
Keywords: random walk in random scenery, local limit theorem, local time, stable process
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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, Volume 49 (2013) no. 2, pp. 506-528. doi : 10.1214/11-AIHP466. http://www.numdam.org/articles/10.1214/11-AIHP466/

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