A functional limit theorem for excited random walks

Andrey Pilipenko

doi:10.1214/17-ecp66

Abstract

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

Highlights

Introduction and resultsLet {X(k), k ≥ 0} be a sequence of Z-valued random variables such that |X(k + 1) − X(k)| = 1, k ≥ 0
We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state
We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time

Summary

Introduction

We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Let {X(k), k ≥ 0} be a sequence of Z-valued random variables such that |X(k + 1) − X(k)| = 1, k ≥ 0. We use Gikhman and Skorokhod result [7] on absolute continuity of the limit process together with the Skorokhod theorem on a single probability space, and invariance principle for the local times of random walks [3].

Results

Conclusion