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Variations of the elephant random walk

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  16 September 2021

Allan Gut  and
Ulrich Stadtmüller Open the ORCID record for Ulrich Stadtmüller [Opens in a new window]
Allan Gut*
Affiliation:
Uppsala University
Ulrich Stadtmüller*
Affiliation:
Ulm University
*
*Postal address: Uppsala University, Department of Mathematics, Box 480, SE-751 06 Uppsala, Sweden. Email address: [email protected]
**Postal address: Ulm University, Department of Number and Probability Theory, 89069 Ulm, Germany. Email address: [email protected]
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Abstract

In the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

Keywords

Elephant random walklaw of large numbersasymptotic (non)normalitymethod of momentsdifference equationMarkov chain

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Secondary: 60F15: Strong theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 805 - 829
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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