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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. randomvariables. We assume here that their distributions belong to the normal domainof 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 orientedlattices for which we obtain similar results.

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