Phase transitions for a unidirectional elephant random walk with a power law memory

Abstract

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter p and the power law exponent β, we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, in the other phase there is a positive probability that the elephant travels an infinite distance and in the third phase the elephant travels an infinite distance with probability 1. For the critical case of the transition from the first phase to the second phase, the proof of our result requires coupling with a multi-type branching process.