# On the center of mass of the elephant random walk

*Bernard Bercu* and
*Lucile Laulin*

*Stochastic Processes and their Applications*, 2021, vol. 133, issue C, 111-128

**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.

**Keywords:** Elephant random walk; Center of mass; Multi-dimensional martingales; Almost sure convergence; Asymptotic normality (search for similar items in EconPapers)

**Date:** 2021

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:spapps:v:133:y:2021:i:c:p:111-128

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**DOI:** 10.1016/j.spa.2020.11.004

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