The Elephant Random Walk

Abstract

The Elephant Random Walk (ERW) is a random walk on Z d , d ≥ 1, with history dependent steps. Due to this history dependence, the walk exhibits certain fascinating properties that are otherwise not present in general random walks. For instance, a phase transition from a diffusive to a superdiffusive regime is observed. We explore the asymptotic properties of ERW and provide alternative proofs of some results. Furthermore, we introduce and study a variation of the ERW with biased memory. We prove a Strong Law and a Central Limit Theorem for such a process for certain parameters. In this ongoing work, we want to examine how such a biased memory affects the behaviour of the walk and how this behaviour differs from a classical ERW.