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 $\alpha \ne 1$ 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.

References in corpus (1)

• Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

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