Functional limit theorems for the multi-dimensional elephant random walk

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