A strong invariance principle for the elephant random walk
Probability
2017-12-18 v1
Abstract
We consider a non-Markovian discrete-time random walk on 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 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.
Cite
@article{arxiv.1707.06905,
title = {A strong invariance principle for the elephant random walk},
author = {Cristian F. Coletti and Renato Gava and Gunter M. Schütz},
journal= {arXiv preprint arXiv:1707.06905},
year = {2017}
}