The Sequential Empirical Process of a Random Walk in Random Scenery

TL;DR

This paper establishes the weak convergence of the sequential empirical process for a random walk in random scenery, revealing a limit process with combined properties of independence and long-range dependence.

Contribution

It introduces the weak convergence of the sequential empirical process for random walk in random scenery, highlighting a new limit process with unique properties.

Findings

Weak convergence of the sequential empirical process is proven.

The limit process exhibits combined features of independence and long-range dependence.

The process shows a new type of behavior with self-similarity and path roughness.

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).