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The elephant random walk in the triangular array setting

Part of: Limit theorems

Published online by Cambridge University Press:  16 January 2025

Rahul Roy Open the ORCID record for Rahul Roy [Opens in a new window] ,
Masato Takei Open the ORCID record for Masato Takei [Opens in a new window]  and
Hideki Tanemura Open the ORCID record for Hideki Tanemura [Opens in a new window]
Rahul Roy*
Affiliation:
Indian Statistical Institute and Indraprastha Institute of Information Technology, Delhi
Masato Takei*
Affiliation:
Yokohama National University
Hideki Tanemura*
Affiliation:
Keio University
*
*Postal address: 7 SJS Sansanwal Marg, New Delhi 110016, India. Email: rahul@isid.ac.in
**Postal address: 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan. Email: takei-masato-fx@ynu.ac.jp
***Postal address: 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan. Email: tanemura@math.keio.ac.jp
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Abstract

Gut and Stadmüller (2021, 2022) initiated the study of the elephant random walk with limited memory. Aguech and El Machkouri (2024) published a paper in which they discuss an extension of the results by Gut and Stadtmüller (2022) for an ‘increasing memory’ version of the elephant random walk without stops. Here we present a formal definition of the process that was hinted at by Gut and Stadtmüller. This definition is based on the triangular array setting. We give a positive answer to the open problem in Gut and Stadtmüller (2022) for the elephant random walk, possibly with stops. We also obtain the central limit theorem for the supercritical case of this model.

Keywords

Random walk with memorylimit theoremsphase transition

MSC classification

Secondary: 60F05: Central limit and other weak theorems

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 3 , September 2025 , pp. 997 - 1009
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

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