Central Limit Theorem for the Elephant Random Walk

TL;DR

This paper investigates the elephant random walk, a non-Markovian process with memory, establishing laws of large numbers and a central limit theorem that applies across different diffusive regimes, including at the phase transition.

Contribution

It proves a central limit theorem for the ERW that holds even at the superdiffusive phase transition, and provides explicit correlation formulas.

Findings

CLT applies in diffusive and superdiffusive regimes

ERW converges to a non-normal distribution in superdiffusive regime

Explicit correlation expressions for increments

Abstract

We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory which exhibits a phase transition from diffusive to superdiffusive behaviour. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.