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 $(\xi_y,y\in{ \mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${ \mathbb Z}^d$ and $\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal basin of attraction of stable distribution of index $\alpha\in(0,2]$ and $\beta\in(0,2]$. When $d=1$ and $\alpha\ne 1$, a functional limit theorem has been established in \cite{ KestenSpitzer} and a local limit theorem in \cite{ BFFN}. In this paper, we establish the convergence of the finite-dimensional distributions and a local limit theorem when $\alpha=d$ (i.e. $\alpha = d=1$ or $\alpha=d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in \cite{ bolthausen} and recently in \cite{ DU} in the particular case where $\beta=2$ (respectively for $\alpha=d=2$ and $\alpha=d=1$).
Categories: math.PR