Open Access
2022 Introducing smooth amnesia to the memory of the Elephant Random Walk
Lucile Laulin
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Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP495


This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.


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Lucile Laulin. "Introducing smooth amnesia to the memory of the Elephant Random Walk." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 1 September 2022; Accepted: 23 October 2022; Published: 2022
First available in Project Euclid: 28 October 2022

MathSciNet: MR4510849
zbMATH: 1500.60019
Digital Object Identifier: 10.1214/22-ECP495

Primary: 60G50
Secondary: 60F17 , 60G42

Keywords: Almost sure convergence , amnesic random walk , asymptotic normality , distributional convergence , elephant random walk , multi-dimensional martingales

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