Introducing smooth amnesia to the memory of the Elephant Random Walk

TL;DR

This paper analyzes the asymptotic behavior of the amnesic elephant random walk using martingale techniques, establishing convergence results and laws in various regimes.

Contribution

It introduces a martingale-based asymptotic analysis of the amnesic elephant random walk, providing new convergence laws and distributional results across different regimes.

Findings

Almost sure convergence in diffusive regimes

Distributional convergence to Gaussian processes

Law of iterated logarithm in critical regime

Abstract

This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.