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. 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 generalization 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 in Schütz and Trimper is that the long-time behavior of the ERW exhibits a phase transition at some critical memory parameter p_c. We investigate the asymptotic behavior 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.

Keywords:

• Elephant random walk
• functional limit theorems
• mathematical physics
• multi-dimensional elephant random walk
• probability theory

Acknowledgements

I would like to express my gratitude to both of my supervisors, Jean Bertoin and Erich Baur, for their patience and valuable input while I was writing this article. Further, I want to thank two anonymous referees for their careful reading of an earlier version of this manuscript and their many insightful comments and suggestions to help improve this article.