A strong invariance principle for the elephant random walk

TL;DR

This paper establishes a strong invariance principle for the elephant random walk, showing it can be approximated by Brownian motion under certain conditions, and derives related limit theorems.

Contribution

It proves a strong invariance principle for the ERW, a non-Markovian process with unbounded memory, in both diffusive and critical regimes, which was previously unestablished.

Findings

ERW is almost surely approximated by Brownian motion under scaling.

Law of iterated logarithm holds for ERW.

Central limit theorem applies to ERW.

Abstract

We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.