On the center of mass of the elephant random walk

TL;DR

This paper studies the long-term behavior of the center of mass in the elephant random walk, revealing convergence properties, limit theorems, and asymptotic normality across different regimes using martingale techniques.

Contribution

It provides the first comprehensive analysis of the asymptotic behavior of the center of mass in the elephant random walk across all regimes.

Findings

Almost sure convergence in diffusive and critical regimes

Law of iterated logarithm established

Asymptotic normality of the center of mass

Abstract

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratric strong law for the center of mass of the elephant random walk. The asymptotic normality of the center of mass, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.