Asymptotic Properties of Generalized Elephant Random Walks

TL;DR

This paper introduces a generalized model of elephant random walks with memory-dependent dynamics, analyzing its asymptotic behavior and phase transitions using stochastic approximation techniques.

Contribution

It proposes a new multidimensional generalized elephant random walk model and extends existing results on stochastic approximation processes.

Findings

Derived asymptotic behavior of the generalized model

Identified phase transition boundary between diffusive and non-diffusive regimes

Extended results on one-dimensional stochastic approximation processes

Abstract

Elephant random walk is a special type of random walk that incorporates the memory of the past to determine its future steps. The probability of this walk taking a particular step (+1 or -1) at a time point, conditioned on the entire history, depends on a linear function of the proportion of steps of that type till that time point. In this work, we consider a generalization of the elephant random walk where we investigate how the dynamics of the random walk will change if we replace this linear function with a generic map satisfying some analytic conditions. We propose a new model, called the multidimensional generalized elephant random walk, that includes several variants of elephant random walk in one and higher dimensions and generalizations thereof. Using tools from the theory of stochastic approximation, we derive the asymptotic behavior of our model leading to newer results on the phase transition boundary between diffusive and non-diffusive regimes. In the process, we extend some results on one-dimensional stochastic approximation process, which can be of independent interest. We also mention a few open problems in this context.