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Limit theorems for one and two-dimensional random walks in random scenery

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 $(ξ_y,y\in{\mathbb Z}^d)$ 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 $ξ_0$ belong to the normal basin of attraction of stable distribution of index $α\in(0,2]$ and $β\in(0,2]$. When $d=1$ and $α\ne 1$, 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 $α=d$ (i.e. $α= d=1$ or $α=d=2$) and $β\in (0,2]$. Let us mention that functional limit theorems have been established in \cite{bolthausen} and recently in \cite{DU} in the particular case where $β=2$ (respectively for $α=d=2$ and $α=d=1$).