Recurrence and transience of multidimensional elephant random walks

2023/09/18 by Qin, Shuo · 6 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2309.09795

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on ℤ^d(d≥ 3) is transient and the expected number of zeros is finite. We also provide some estimates on the rate of escape. In dimensions d= 1, 2, we prove that phase transitions between recurrence and transience occur at p=(2d+1)/(4d). Let S be an elephant random walk with parameter p. For p ≤ 3/4, we provide a Berry-Esseen type bound for properly normalized S_n. For p>3/4, the distribution of lim_{n→ ∞} S_n/n^2p-1 will be studied.

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