A Hybrid Model of Elephant and Moran Random Walks: Exact Distribution and Symmetry Properties

This work introduces a hybrid memory-based random walk model that combines the Elephant Random Walk with a modified Moran Random Walk. The model introduces a sequence of independent and identically distributed random variables with mean 1, representing step sizes. A particle starts at the origin and moves upward with probability r or remains stationary with probability $1-r$. From the second step onward, the particle decides its next action based on its previous movement, repeating it with probability p or taking the opposite action with probability $1-p$. The novelty of our approach lies in integrating a short-memory mechanism with variable step sizes, which allows us to derive exact distributions, recurrence relations, and central limit theorems. Our main contributions include (i) establishing explicit expressions for the moment-generating function and the exact distribution of the process, (ii) analyzing the number of stops through a symmetry phenomenon between repetition and inversion, and (iii) providing asymptotic results supported by simulations.