On Elephant Random Walk with Random Memory

TL;DR

The paper studies the impact of random memory on the elephant random walk by introducing a time-changed ERW where memory is sampled via random memory sets. It develops conditional distribution results for increments and displacement and then derives recursive and explicit formulas for the mean increments and the mean displacement, using combinatorial index sets $\Phi^j_n$ and $\Theta^j_n$ and parameters $\alpha=2p-1$ and $\beta=\mathbb{E}(X_1)$. It shows that the means are polynomials in $\alpha$ of degree $n$, providing a generalization of the standard ERW and a concrete toolkit for analyzing memory-driven diffusion in one dimension. These results help quantify how random memory affects diffusion and may inform models of anomalous diffusion where memory is random.

Abstract

In this paper, we introduce the elephant random walk (ERW) with memory consisting of randomly selected steps from its history. It is a time-changed variant of the standard elephant random walk with memory consisting of its full history. At each time point, the time changing component is the composition of two uniformly distributed independent random variables with support over all the past steps. Several conditional distributional properties including the conditional mean increments and conditional displacement of ERW with random memory are obtained. Using these conditional results, we derive the recursive and explicit expressions for the mean increments and mean displacement of the walk.