Elephant Random Walk with multiple extractions

TL;DR

This paper extends the Elephant Random Walk model to cases with multiple previous steps, revealing complex dependencies on initial conditions and memory parameters, and analyzing convergence behaviors using urn model analogies.

Contribution

It introduces a generalized model for the Elephant Random Walk with multiple steps, and develops a novel approach using urn models to analyze its properties.

Findings

For k=1, the model is exactly solvable.

For k>2, the model exhibits critical dependence on initial conditions.

Regions of convergence have entropy sub-linear in steps.

Abstract

Consider a generalized Elephant Random Walk in which the step is chosen by selecting $k$ previous steps with $k$ odd and then going in the majority direction with a probability $p$ and in the opposite direction otherwise. In the $k=1$ case the model is the original one and could be resolved exactly by analogy with Friedman's urn. However the analogy cannot be extended to the $k>2$ case already. In this paper we show how to treat the model for each $k$ by analogy with the more general urn model of Hill, Lane and Sudderth. Interestingly for $k>2$ we found a critical dependence from the initial conditions beyond a certain values of the memory parameter $p$, and regions of convergence with entropy that is sub-linear in the number of steps.