Recurrence and transience of multidimensional elephant random walks

TL;DR

This paper investigates the recurrence and transience properties of multidimensional elephant random walks, proving conjectures, analyzing phase transitions, and providing distributional limits and bounds based on the parameter p.

Contribution

It proves a conjecture on transience in higher dimensions, characterizes phase transitions in lower dimensions, and derives distributional and convergence results for different parameter regimes.

Findings

Multi-dimensional elephant random walk is transient for dimensions d≥3.

Phase transitions between recurrence and transience occur at p=(2d+1)/(4d) for d=1,2.

Distributional limits and Berry-Esseen bounds are established depending on p.

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on $\mathbb{Z}^d$($d\geq 3$) is transient and the expected number of zeros is finite. We also provide some estimates on the rate of escape. In dimensions $d= 1, 2$, we prove that phase transitions between recurrence and transience occur at $p=(2d+1)/(4d)$. Let $S$ be an elephant random walk with parameter $p$. For $p \leq 3/4$, we provide a Berry-Esseen type bound for properly normalized $S_n$. For $p>3/4$, the distribution of $\lim_{n\to \infty} S_n/n^{2p-1}$ will be studied.