A martingale approach for the elephant random walk

TL;DR

This paper uses a martingale approach to analyze the asymptotic behavior of the one-dimensional elephant random walk, revealing new results on convergence and normality across different memory regimes.

Contribution

It introduces a martingale-based method to refine understanding of ERW's asymptotics, including convergence and normality, across various regimes.

Findings

Almost sure convergence in different regimes

Asymptotic normality established

Behavior varies with memory parameter p

Abstract

The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$ which lies between zero and one. This behavior is totally different in the diffusive regime $0 \leq p <3/4$, the critical regime $p=3/4$, and the superdiffusive regime $3/4<p \leq 1$. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.