Limit theorem for random walk in weakly dependent random scenery

Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $(\Sigma_n)_{n\geq 0}$ where $$\Sigma_n=\sum_{k=0}^n \xi_{S_k}, n\in\mathbb{N}.$$ Under a weak dependence assumption on the scenery $\xi$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).