The sequential empirical process of a random walk in random scenery

A random walk in random scenery (Yn)n∈N(Yn)n∈N is given by Yn=ξSnYn=ξSn for a random walk (Sn)n∈N(Sn)n∈N and i.i.d. random variables (ξn)n∈Z(ξn)n∈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).