Limit theorem for random walk in weakly dependent random scenery

Nadine Guillotin-Plantard (1) , Clémentine Prieur (2, 3)

1 ICJ - Institut Camille Jordan
2 LSP - Laboratoire de Statistique et Probabilités
3 MOISE - Modelling, Observations, Identification for Environmental Sciences

Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $\xi=(\xi_{i})_{i\in\mathbb{Z}}$ 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 $(\Sigma_n)_{n\geq 0}$ where $$\Sigma_n=\sum_{k=0}^n \xi_{S_k},\ n\in\mathbb{N}.$$ Under a weak dependence assumption on the scenery $\xi$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).

Mots clés: Random walks; Random scenery; Weak dependence; Limit theorem; Local time

Domaines: Probabilités [math.PR]

Nadine Guillotin-Plantard, Clémentine Prieur. Limit theorem for random walk in weakly dependent random scenery. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2010, 46 (4), pp.1178-1194. ⟨10.1214/09-AIHP353⟩. ⟨hal-00303698⟩