New insights on the reinforced Elephant Random Walk using a martingale approach

TL;DR

This paper provides a comprehensive asymptotic analysis of the reinforced elephant random walk (RERW) across different regimes using martingale techniques, revealing convergence behaviors and Gaussian process limits.

Contribution

It introduces a martingale-based framework to analyze RERW, establishing new convergence results and distributional limits in various regimes.

Findings

Almost sure convergence in diffusive and critical regimes

Distributional convergence to Gaussian processes

Mean square convergence in superdiffusive regime

Abstract

This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization.