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arXiv · 1707.06905

A strong invariance principle for the elephant random walk

Cristian F. Coletti·Renato Gava·Gunter M. Schütz
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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.

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Cristian F. Coletti, Renato Gava, Gunter M. Schütz. 2017-07-21. A strong invariance principle for the elephant random walk. https://doi.org/10.1088/1742-5468/aa9680

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