On the center of mass of the elephant random walk

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:133

Article

Bercu, Bernard¹  Laulin, Lucile¹ 

[1] Univ Bordeaux, Inst Math Bordeaux, UMR 5251, 351 Cours Liberat, F-33405 Talence, France

关键词: Elephant random walk;    Center of mass;    Multi-dimensional martingales;    Almost sure convergence;    Asymptotic normality;   

DOI  :  10.1016/j.spa.2020.11.004

来源: Elsevier

【 摘 要 】

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the center of mass of the elephant random walk. The asymptotic normality, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free