Renewal theorems for random walks in random scenery

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 suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal domain of attraction of strictly stable distributions with index $\alpha\in[1,2]$ and $\beta\in(0,2)$ respectively. We are interested in the asymptotic behaviour as $|a|$ goes to infinity of quantities of the form $\sum_{n\ge 1}{\mathbb E}[h(Z_n-a)]$ (when $(Z_n)n$ is transient) or $\sum{n\ge 1}{\mathbb E}[h(Z_n)-h(Z_n-a)]$ (when $(Z_n)_n$ is recurrent) where $h$ is some complex-valued function defined on $\mathbb{R}$ or $\mathbb{Z}$.