Functional limit theorems for the Multi-dimensional Elephant Random Walk

Abstract

In this article we shall derive functional limit theorems for the multi-dimensional elephant random walk (MERW) and thus extend the results provided for the one-dimensional marginal by Bercu and Laulin (2019). The MERW is a non-Markovian discrete time-random walk on ${\mathbb{Z}}^d$ which has a complete memory of its whole past, in allusion to the traditional saying that an elephant never forgets. As the name suggests, the MERW is a $d$-dimensional generalisation of the elephant random walk (ERW), the latter was first introduced by Schütz and Trimper in 2004. We measure the influence of the elephant's memory by a so-called memory parameter $p$ between zero and one. A striking feature that has been observed by Schütz and Trimper is that the long-time behaviour of the ERW exhibits a phase transition at some critical memory parameter $p_c$. We investigate the asymptotic behaviour of the MERW in all memory regimes by exploiting a connection between the MERW and Pólya urns, following similar ideas as in the work by Baur and Bertoin for the ERW.

This new version respects two referees' corrections and has been accepted for publication in Stochastic Models