Quenched central limit theorems for random walks in random scenery

Random walks in random scenery are processes defined by Z n : = ∑ k = 1 n ω S k where S : = ( S k , k ≥ 0 ) is a random walk evolving in Z d and ω : = ( ω x , x ∈ Z d ) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk S and the random scenery ω , almost surely with respect to ω , the correctly renormalized sequence ( Z n ) n ≥ 1 is proved to converge in distribution to a centered Gaussian law with explicit variance.