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

Rafik Aguech; Rafik Aguech

doi:10.3934/math.2024865

AIMS Mathematics

2024, Volume 9, Issue 7: 17784-17794. doi: 10.3934/math.2024865

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Research article

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

Rafik Aguech ^,

Department of Statistics and Operation research, King Saud University, Saudi Arabia

Received: 28 March 2024 Revised: 11 May 2024 Accepted: 13 May 2024 Published: 24 May 2024
MSC : 60F05, 60F25, 60G42, 60G50

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 $.
- elephant random walk,
- central limit theorem,
- asymptotic normality,
- phase transition,
- martingale theory
Citation: Rafik Aguech. On the central limit theorem for the elephant random walk with gradually increasing memory and random step size[J]. AIMS Mathematics, 2024, 9(7): 17784-17794. doi: 10.3934/math.2024865

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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 $.

References

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