On Elephant Random Walk with Delayed Amnesia

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 walk across different parameter regimes.

In the diffusive and critical regimes, we establish almost sure convergence results, laws of iterated logarithm, asymptotic normality of the walk, and the growth rate of mean square displacement.

In the superdiffusive regime, we prove an almost sure convergence result and obtain the corresponding mean square displacement rate for the walk.

Also, we study some almost sure convergence results for its center of mass.

Later, we extend the model by incorporating random step sizes and obtain asymptotic results for it.