Soit S=(Sk)k≥0 une marche aléatoire sur ℤ et ξ=(ξi)i∈ℤ une suite stationnaire de variables aléatoires centrées, indépendante de S. Nous considérons une marche aléatoire en scène aléatoire définie par la suite de variables aléatoires (Un)n≥0=(∑k=0nξSk)n≥0. Sous une hypothèse de dépendance faible portant sur la scène ξ, nous montrons un théorème de la limite centrale fonctionnel généralisant le théorème de Kesten et Spitzer [Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25].
Let S=(Sk)k≥0 be a random walk on ℤ and ξ=(ξi)i∈ℤ a stationary random sequence of centered random variables, independent of S. We consider a random walk in random scenery that is the sequence of random variables (Un)n≥0, where Un=∑k=0nξSk, n∈ℕ. Under a weak dependence assumption on the scenery ξ we prove a functional limit theorem generalizing Kesten and Spitzer's [Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25] theorem.
Mots clés : random walks, random scenery, weak dependence, limit theorem, local time
@article{AIHPB_2010__46_4_1178_0, author = {Guillotin-Plantard, Nadine and Prieur, Cl\'ementine}, title = {Limit theorem for random walk in weakly dependent random scenery}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1178--1194}, publisher = {Gauthier-Villars}, volume = {46}, number = {4}, year = {2010}, doi = {10.1214/09-AIHP353}, mrnumber = {2744890}, zbl = {1219.60022}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP353/} }
TY - JOUR AU - Guillotin-Plantard, Nadine AU - Prieur, Clémentine TI - Limit theorem for random walk in weakly dependent random scenery JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 1178 EP - 1194 VL - 46 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP353/ DO - 10.1214/09-AIHP353 LA - en ID - AIHPB_2010__46_4_1178_0 ER -
%0 Journal Article %A Guillotin-Plantard, Nadine %A Prieur, Clémentine %T Limit theorem for random walk in weakly dependent random scenery %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 1178-1194 %V 46 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP353/ %R 10.1214/09-AIHP353 %G en %F AIHPB_2010__46_4_1178_0
Guillotin-Plantard, Nadine; Prieur, Clémentine. Limit theorem for random walk in weakly dependent random scenery. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1178-1194. doi : 10.1214/09-AIHP353. http://www.numdam.org/articles/10.1214/09-AIHP353/
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