Analysis of the smoothly amnesia-reinforced multidimensional elephant random walk

TL;DR

This paper investigates the properties of a multidimensional elephant random walk with amnesia, establishing its scaling limits, convergence behaviors, and deviation properties across different regimes.

Contribution

It introduces the smoothly amnesia-reinforced multidimensional elephant random walk and analyzes its scaling limits, convergence types, and deviation behaviors in detail.

Findings

Existence of scaling limits in diffusive, critical, and superdiffusive regimes.

Almost sure convergence in all three regimes.

Mean square convergence to a non-Gaussian variable in the superdiffusive regime.

Abstract

In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical and superdiffusive regimes. We also establish the almost sure convergence in all of the three regimes. The quadratic strong law is displayed in the diffusive regime as well as in the critical regime. The mean square convergence towards a non-Gaussian random variable is established in the superdiffusive regime. Similar results for the barycenter process are also derived. Finally, the last two Sections are devoted to a discussion of the convergence velocity of the mean square displacement and the Cram{\'e}r moderate deviations.