Quenched Central Limit Theorems for Random Walks in Random Scenery

Random walks in random scenery are processes defined by $$Z_n:=\sum _{k=1}^n{\omega }_{S_k}$$ where $S:=(S_k,k\ge 0)$ is a random walk evolving in ${\mathbb{Z}}^d$ and $\omega :=({\omega }_x,x\in {\mathbb{Z}}^d)$ is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk $S$ and the random scenery $\omega$, almost surely with respect to $\omega$, the correctly renormalized sequence $(Z_n{)}_{n\ge 1}$ is proved to converge in distribution to a centered Gaussian law with explicit variance.