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 $(Z_n)_{ n\geq 1}$ is proved to converge in distribution to a centered Gaussian law with explicit variance.
Categories: math.PR