On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution

TL;DR

This paper studies the asymptotic behavior of the elephant random walk with stops, revealing convergence to Mittag-Leffler distributions and establishing conditions for normality and nondegenerate limits across different regimes.

Contribution

It introduces the ERWS model allowing the elephant to stay put and provides a detailed asymptotic analysis with new convergence results and distributional limits.

Findings

Number of ones converges to Mittag-Leffler distribution.

Almost sure convergence in diffusive and critical regimes.

Asymptotic normality after self-normalization.

Abstract

The aim of this paper is to investigate the asymptotic behavior of the so-called elephant random walk with stops (ERWS). In contrast with the standard elephant random walk, the elephant is allowed to be lazy by staying on his own position. We prove that the number of ones of the ERWS, properly normalized, converges almost surely to a Mittag-Leffler distribution. It allows us to carry out a sharp analysis on the asymptotic behavior of the ERWS. In the diffusive and critical regimes, we establish the almost sure convergence of the ERWS. We also show that it is necessary to self-normalized the position of the ERWS by the random number of ones in order to prove the asymptotic normality. In the superdiffusive regime, we establish the almost sure convergence of the ERWS, properly normalized, to a nondegenerate random variable. Moreover, we also show that the fluctuation of the ERWS around its limiting random variable is still Gaussian.