On Elephant Random Walk with Delayed Amnesia

Abstract

In this paper, we introduce a modified elephant random walk that exhibits a transition from a uniform memory mechanism to a selective amnesic memory mechanism. Using a vector martingale approach, we study the asymptotic behaviour of the model across different parameter regimes. In the diffusive and critical regimes, we establish almost sure convergence, laws of the iterated logarithm, asymptotic normality of the walk, and the asymptotic rate of the mean square displacement. In the superdiffusive regime, we prove an almost sure convergence result and derive the corresponding mean square displacement rate. Also, we study the asymptotic behaviour of the center of mass associated with the walk. Later, we extend the model by incorporating random step sizes and obtain similar asymptotic results.