Limit theorem for random walk in weakly dependent random scenery
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 1178-1194

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.

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].

DOI : https://doi.org/10.1214/09-AIHP353
Classification:  60F05,  60G50,  62D05,  37C30,  37E05
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {1178-1194},
     doi = {10.1214/09-AIHP353},
     zbl = {1219.60022},
     mrnumber = {2744890},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 46 (2010) no. 4, pp. 1178-1194. doi : 10.1214/09-AIHP353. http://www.numdam.org/item/AIHPB_2010__46_4_1178_0/

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