Elephant Random Walks and their connection to P\'olya-type urns

TL;DR

This paper explores the connection between Elephant Random Walks and Pólya urns, providing explicit solutions across all memory regimes and extending applicability to higher dimensions and related models.

Contribution

It establishes a link between ERWs and urn models, enabling explicit solutions and analysis of ERWs in various memory regimes and dimensions.

Findings

Explicit solutions for ERWs in all memory regimes

Extension of methods to higher-dimensional ERWs

Applicability to related memory-influenced models

Abstract

In this paper, we explain the connection between the Elephant Random Walk (ERW) and an urn model \`a la P\'olya and derive functional limit theorems for the former. The ERW model was introduced by Sch\"utz and Trimper [2004] to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a parameter $p$ between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of $p$. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.