Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

2023/02/13 by Jérôme Dedecker, Dedecker, Jérôme, Xiequan Fan +5 · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2302.06311

openalex publication_date 2023/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

Cited by

• A central limit theorem for unbalanced step-reinforced random walks
• Elephant random walks with multiple extractions and general reinforcement functions
• Elephant random walk with polynomially decaying steps

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