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 $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ 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 $α\in (0,2]$ and $β\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $α\neq 1$ and as $n\to \infty$, of $n^{-δ}Z_n$, for some suitable $δ>0$ depending on $α$ and $β$. Here we are interested in the convergence, as $n\to \infty$, of $n^δ{\mathbb P}(Z_n=\lfloor n^δ x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.