  A random walk in random scenery $(Y_n)_{n\in\mathbb{N}}$ is given by
$Y_n=\xi_{S_n}$ for a random walk $(S_n)_{n\in\mathbb{N}}$ and iid random
variables $(\xi_n)_{n\in\mathbb{Z}}$. In this paper, we will show the weak
convergence of the sequential empirical process, i.e. the centered and rescaled
empirical distribution function. The limit process shows a new type of
behavior, combining properties of the limit in the independent case (roughness
of the paths) and in the long range dependent case (self-similarity).
