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

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 Author Castell, Fabienne ♦ Guillotin-Plantard, Nadine ♦ Pène, Françoise ♦ Schapira, Bruno Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-09 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Probability ♦ 60F05 ♦ 60G52 ♦ math 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. Educational Use Research Learning Resource Type Article