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On the center of mass of the elephant random walk

Author

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  • Bercu, Bernard
  • Laulin, Lucile

Abstract

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the center of mass of the elephant random walk. The asymptotic normality, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.

Suggested Citation

  • Bercu, Bernard & Laulin, Lucile, 2021. "On the center of mass of the elephant random walk," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 111-128.
    • Handle: RePEc:eee:spapps:v:133:y:2021:i:c:p:111-128
    DOI: 10.1016/j.spa.2020.11.004
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    References listed on IDEAS

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    1. Lo, Chak Hei & Wade, Andrew R., 2019. "On the centre of mass of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4663-4686.
    2. Grill, Karl, 1988. "On the average of a random walk," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 357-361, April.
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    2. Lo, Chak Hei & Wade, Andrew R., 2019. "On the centre of mass of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4663-4686.

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