On the multidimensional elephant random walk with stops

TL;DR

This paper analyzes the asymptotic behavior of the multidimensional elephant random walk with stops, extending previous one-dimensional results and exploring convergence properties across different regimes.

Contribution

It introduces the multidimensional elephant random walk with stops and proves its asymptotic convergence properties, extending known results from the one-dimensional case.

Findings

Gram matrix converges almost surely to a deterministic product and Mittag-Leffler distribution.

Almost sure convergence in diffusive and critical regimes.

Convergence to a nondegenerate random vector in superdiffusive regime.

Abstract

The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the product of a deterministic matrix, related to the axes on which the MERWS moves uniformly, and a Mittag-Leffler distribution. It allows us to extend all the results previously established for the one-dimensional elephant random walk with stops. More precisely, in the diffusive and critical regimes, we prove the almost sure convergence of the MERWS. In the superdiffusive regime, we establish the almost sure convergence of the MERWS, properly normalized, to a nondegenerate random vector. We also study the self-normalized asymptotic normality of the MERWS.