Functional Limits and Separation Times for Two Interacting Elephant Random Walks

Abstract

We establish functional scaling limits and study first separation times for the interacting two elephant model studied by Aguech and Qin. In the joint diffusive regime, we give a direct martingale proof of convergence to a two-dimensional continuous Gaussian process represented by a matrix-kernel analogue of the noise-reinforced Brownian motion. We then investigate the difference process, whose diffusive scaling persists in the symmetric case even when the joint walk is critical or superdiffusive. For the first separation time of the two walks, we establish convergence in distribution to the first exit time of the limiting Gaussian process, together with convergence of all positive moments under diffusive scaling. We also obtain monotonicity results for the limiting exit time by combining explicit covariance identities with Anderson's inequality.