A central limit theorem for two-dimensional random walks in random sceneries

Abstract

Let $S_n$, $n\in\bold N$, be a recurrent random walk on ${\bold Z}^2$ $(S_0=0)$ and let $\xi(\alpha)$, $\alpha\in{\bold Z}^2$, be i.i.d. $\bold R$-valued centered random variables. It is shown that $\sum^n_{i=1}\xi(S_i)/ \sqrt{n\log n}$ satisfies a central limit theorem. A functional version is also presented.