On the central limit theorem for the elephant random walk with gradually increasing memory and random step size

• Published in: AIMS mathematics Vol. 9; no. 7; pp. 17784 - 17794
• Main Author: Aguech, Rafik
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2024
• Subjects: asymptotic normality; central limit theorem; elephant random walk; martingale theory; phase transition
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024865

In this paper, we investigate an extended version of the elephant random walk model. Unlike the traditional approach where step sizes remain constant, our model introduces a novel feature: step sizes are generated as a sequence of positive independent and identically distributed random variables, and the step of the walker at time $ n+1 $ depends only on the steps of the walker between times $ 1, ..., m_n $, where $ (m_n)_{n\geqslant 1} $ is a sequence of positive integers growing to infinity as $ n $ goes to infinity. Our main results deal with the validity of the central limit theorem for this new variation of the standard ERW model introduced by Schütz and Trimper in $ 2004 $.