Recurrence and transience of multidimensional elephant random walks

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on \(\mathbb{Z}^d\)(\(d\geq 3\)) 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 \(p \leq 3/4\), we provide a Berry-Esseen type bound for properly normalized \(S_n\). For p>3/4, the distribution of \(\lim_{n\to \infty} S_n/n^{2p-1}\) will be studied.