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

Abstract : Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ 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 $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.
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Submitted on : Tuesday, February 9, 2010 - 4:07:28 PM
Last modification on : Monday, October 11, 2021 - 2:22:07 PM
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Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pene, Bruno Schapira. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Annals of Probability, Institute of Mathematical Statistics, 2011, pp.Vol. 39, No 6, 2079--2118. ⟨10.1214/10-AOP606⟩. ⟨hal-00455154⟩

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