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The elephant random walk with gradually increasing memory

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  • Gut, Allan
  • Stadtmüller, Ulrich

Abstract

In the simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk (ERW), which was introduced by Schütz and Trimper (2004), the next step always depends on the whole path so far. Various authors have studied further properties of the ERW. In Gut and Stadtmüller (2021b) we studied the case when the Elephant remembers only a finite part of the first or last steps. In both cases there was no separation into two different regimes as in the classical ERW. We also posed the question about what happens if she remembers a gradually increasing past. This paper will give some answers to that question. We also discuss related questions for ERW:s with delays.

Suggested Citation

  • Gut, Allan & Stadtmüller, Ulrich, 2022. "The elephant random walk with gradually increasing memory," Statistics & Probability Letters, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:stapro:v:189:y:2022:i:c:s0167715222001432
    DOI: 10.1016/j.spl.2022.109598
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    References listed on IDEAS

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    1. C.C. Heyde, 2004. "Asymptotics and Criticality for a Correlated Bernoulli Process," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 46(1), pages 53-57, March.
    2. James, Barry & James, Kang & Qi, Yongcheng, 2008. "Limit theorems for correlated Bernoulli random variables," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2339-2345, October.
    3. Gut, Allan & Stadtmüller, Ulrich, 2021. "The number of zeros in Elephant random walks with delays," Statistics & Probability Letters, Elsevier, vol. 174(C).
    4. Moura, Thiago R.S. & Viswanathan, G.M. & da Silva, M.A.A. & Cressoni, J.C. & da Silva, L.R., 2016. "Transient superdiffusion in random walks with a q-exponentially decaying memory profile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 259-263.
    5. da Silva, M.A.A. & Cressoni, J.C. & Viswanathan, G.M., 2006. "Discrete-time non-Markovian random walks: The effect of memory limitations on scaling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 364(C), pages 70-78.
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