Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theorem for random walk in weakly dependent random scenery

Nadine Guillotin-Plantard and Clémentine Prieur

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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].

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 1178-1194.

First available in Project Euclid: 4 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 62D05: Sampling theory, sample surveys
Secondary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

Random walks Random scenery Weak dependence Limit theorem Local time


Guillotin-Plantard, Nadine; Prieur, Clémentine. Limit theorem for random walk in weakly dependent random scenery. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1178--1194. doi:10.1214/09-AIHP353.

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