The Sequential Empirical Process of a Random Walk in Random Scenery


A random walk in random scenery (Yn)n∈N(Y_n)_{n\in\mathbb{N}} is given by Yn=ξSnY_n=\xi_{S_n} for a random walk (Sn)n∈N(S_n)_{n\in\mathbb{N}} and iid random variables (ξn)n∈Z(\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)

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