Limit theorems for amnesic elephant Random walks with random increment sizes

Abstract

We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.