Les promenades aléatoires en paysage aléatoire sont des processus définis par , où et sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans et respectivement. Nous supposons que les lois de et appartiennent au domaine d’attraction normal de lois stables d’indice et . Quand et , 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 (i.e. ou ) et . 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 (respectivement pour et ).
Random walks in random scenery are processes defined by , where and are two independent sequences of i.i.d. random variables with values in and respectively. We suppose that the distributions of and belong to the normal basin of attraction of stable distribution of index and . When and , 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 (i.e. or ) and . 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 (respectively for and ).
Mots clés : random walk in random scenery, local limit theorem, local time, stable process
@article{AIHPB_2013__49_2_506_0, author = {Castell, Fabienne and Guillotin-Plantard, Nadine and P\`ene, Fran\c{c}oise}, title = {Limit theorems for one and two-dimensional random walks in random scenery}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {506--528}, publisher = {Gauthier-Villars}, volume = {49}, number = {2}, year = {2013}, doi = {10.1214/11-AIHP466}, mrnumber = {3088379}, zbl = {1278.60046}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP466/} }
TY - JOUR AU - Castell, Fabienne AU - Guillotin-Plantard, Nadine AU - Pène, Françoise TI - Limit theorems for one and two-dimensional random walks in random scenery JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 506 EP - 528 VL - 49 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP466/ DO - 10.1214/11-AIHP466 LA - en ID - AIHPB_2013__49_2_506_0 ER -
%0 Journal Article %A Castell, Fabienne %A Guillotin-Plantard, Nadine %A Pène, Françoise %T Limit theorems for one and two-dimensional random walks in random scenery %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 506-528 %V 49 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP466/ %R 10.1214/11-AIHP466 %G en %F 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/articles/10.1214/11-AIHP466/
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