A strong invariance principle for the elephant random walk

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.

References in corpus (3)

• A martingale approach for the elephant random walk
• Current fluctuations in stochastic systems with long-range memory
• Exact moments in a continuous time random walk with complete memory of its history