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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Counting the zeros of an elephant random walk
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by Jean Bertoin PDF
Trans. Amer. Math. Soc. 375 (2022), 5539-5560 Request permission


We study how memory impacts passages at the origin for a so-called elephant random walk in the diffusive regime. We observe that the number of zeros always grows asymptotically like the square root of the time, despite the fact that, depending on the memory parameter, first return times to $0$ may have a finite expectation or a fat tail with exponent less than $1/2$. We resolve this apparent paradox by recasting the questions in the framework of scaling limits for Markov chains and self-similar Markov processes.
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Additional Information
  • Jean Bertoin
  • Affiliation: Institute of Mathematics, University of Zurich, Switzerland
  • MR Author ID: 237984
  • ORCID: 0000-0002-0073-0439
  • Received by editor(s): May 20, 2021
  • Received by editor(s) in revised form: December 13, 2021, and December 14, 2021
  • Published electronically: March 31, 2022
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 5539-5560
  • MSC (2020): Primary 60J10, 60J55, 82C41, 60G42
  • DOI:
  • MathSciNet review: 4469228