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.
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BibTeX
@misc{aguech2026functional,
title = {{Functional Limits and Separation Times for Two Interacting Elephant Random Walks}},
author = {Rafik Aguech and Shuhei Shibata and Tomoyuki Shirai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39455},
eprint = {2609.39455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39455},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Functional Limits and Separation Times for Two Interacting Elephant Random Walks
AU - Rafik Aguech
AU - Shuhei Shibata
AU - Tomoyuki Shirai
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39455
UR - https://arxiv.org/abs/2609.39455
AB - 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.
ER -