Asymptotic Analysis of the Elephant Random Walk

TL;DR

This paper investigates the long-term behavior of the elephant random walk, a model with long-range memory, establishing conditions for recurrence, transience, and analyzing its mean field limit.

Contribution

It provides a comprehensive asymptotic analysis of the elephant random walk, including recurrence, transience, and mean field limit under the Poisson Hypothesis.

Findings

Proved recurrence and positive recurrence conditions.

Established the transience regime.

Derived an upper bound for the expected distance from the origin.

Abstract

In this work we study asymptotic properties of a long range memory random walk known as elephant random walk. First we prove recurrence and positive recurrence for the elephant random walk. Then, we establish the transience regime of the model. Finally, under the Poisson Hypothesis, we study the replica mean field limit for this random walk and we obtain an upper bound for the expected distance of the walker from the origin.