Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes

Abstract

In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.