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November 2011 A local limit theorem for random walks in random scenery and on randomly oriented lattices
Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène, Bruno Schapira
Ann. Probab. 39(6): 2079-2118 (November 2011). DOI: 10.1214/10-AOP606


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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Fabienne Castell. Nadine Guillotin-Plantard. Françoise Pène. Bruno Schapira. "A local limit theorem for random walks in random scenery and on randomly oriented lattices." Ann. Probab. 39 (6) 2079 - 2118, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1238.60028
MathSciNet: MR2932665
Digital Object Identifier: 10.1214/10-AOP606

Primary: 60F05 , 60G52

Keywords: local limit theorem , Random walk in random scenery , random walk on randomly oriented lattices , Stable process

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 6 • November 2011
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