A functional limit theorem for for excited random walks

TL;DR

This paper proves that excited random walks, when scaled appropriately, converge to an excited Brownian motion described by an SDE with local time-dependent drift, using a novel Radon-Nikodym density approach.

Contribution

It introduces a new method based on Radon-Nikodym densities to establish the limit theorem for excited random walks, extending previous results.

Findings

ERW converges to excited Brownian motion under scaling

New proof technique using Radon-Nikodym densities

Extension of previous results by Raimond and Schapira

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 of investigations based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.