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 that step till that time point. In this work, we investigate how the dynamics of the random walk will change if we replace this linear function by a generic map satisfying some analytic conditions. We also propose a new model, called the multidimensional generalized elephant random walk, that incorporates several variants of elephant random walk in one and higher dimensions and their 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 superdiffusive regimes. We also mention a few open problems in this context.