The Sequential Empirical Process of a Random Walk in Random Scenery

Abstract

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).