Asymptotic Properties of Generalized Elephant Random Walks

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.

39 pages; Section 4 has been revised and detailed proofs to Theorems 4.8 and 4.10 are provided, which are new. Additional examples and remarks are added to facilitate understanding