Renewal theorems for random walks in random scenery

Random walks in random scenery are processes defined by $Z_n:={\sum }_{k=1}^n{\xi }_{X_1+\mathrm{.}\mathrm{.}\mathrm{.}+X_k}$, 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 $\mathrm{\mid }a\mathrm{\mid }$ 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 $\mathbb{Z}$.