Counting the zeros of an elephant random walk

TL;DR

This paper investigates how memory influences the zero-crossing behavior of an elephant random walk, revealing that zeros grow proportionally to the square root of time regardless of return time distributions, through scaling limit analysis.

Contribution

It introduces a novel analysis of the zero-counting behavior in elephant random walks using scaling limits, resolving paradoxes related to return time distributions.

Findings

Zeros grow asymptotically as the square root of time.

Return times to zero can have finite expectation or heavy tails.

Scaling limits explain the zero growth behavior.

Abstract

We study how memory impacts passages at the origin for a so-called elephant random walk in the diffusive regime. We observe that the number of zeros always grows asymptotically like the square root of the time, despite the fact that, depending on the memory parameter, first return times to $0$ may have a finite expectation or a fat tail with exponent less than $1/2$. We resolve this apparent paradox by recasting the questions in the framework of scaling limits for Markov chains and self-similar Markov processes.