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 Sch\"utz 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 Sch\"utz 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 P\'olya urns, following similar ideas as in the work by Baur and Bertoin for the ERW.