The Annals of Probability

A local limit theorem for random walks in random scenery and on randomly oriented lattices

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène, and Bruno Schapira

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Random walks in random scenery are processes defined by Zn := ∑k=1nξX1+⋯+Xk, where (Xk, k ≥ 1) and (ξy, y ∈ ℤ) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2], respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α ≠ 1 and as n → ∞, of nδZn, for some suitable δ > 0 depending on α and β. Here, we are interested in the convergence, as n → ∞, of nδℙ(Zn = ⌊nδx⌋), when x ∈ ℝ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

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Ann. Probab., Volume 39, Number 6 (2011), 2079-2118.

First available in Project Euclid: 17 November 2011

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Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes

Random walk in random scenery random walk on randomly oriented lattices local limit theorem stable process


Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise; Schapira, Bruno. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Ann. Probab. 39 (2011), no. 6, 2079--2118. doi:10.1214/10-AOP606.

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