A limit theorem for a random walk in a stationary scenery coming from a hyperbolic dynamical system

In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(\xi\_k:=f(T^k(.)))\_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S\_n)\_n$ be a simple symmetric random walk on $Z$ independent of $(\xi\_k)\_k$. We give examples of partially hyperbolic dynamical systems and of functions $f$ such that $n^{-3/4}(\xi(S\_1)+...+\xi(S\_k))$ converges in distribution as $n$ goes to infinity.