How often does a critical elephant random walk return to origin

TL;DR

This paper investigates the long-term behavior of the critical elephant random walk, focusing on how frequently it returns to the origin and the distribution of its first return time.

Contribution

It provides new insights into the asymptotic behavior and tail estimates of the critical elephant random walk's return times.

Findings

Most zeros occur shortly before the last zero passage

Derived tail estimate for the first return time

Characterized the asymptotic behavior of origin visits

Abstract

We study the asymptotic behaviour of the number of times the elephant random walk in the critical regime visits the origin. Our result entails that most zeros of the critical elephant random walk occur shortly before its last passage time at zero. Additionally, we derive the tail estimate of the first return time of the random walk.