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

Abstract

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 $.

Keywords

• elephant random walk
• central limit theorem
• asymptotic normality
• phase transition
• martingale theory