Recurrence and transience of multidimensional elephant random walks

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on ${\mathbb{Z}}^d$($d\ge 3$) 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 $p=(2d+1)\mathrm{/}(4d)$. Let $S$ be an elephant random walk with parameter $p$. For $p\le 3\mathrm{/}4$, we provide a Berry-Esseen type bound for properly normalized $S_n$. For $p>3\mathrm{/}4$, the distribution of ${\lim }_{n\to \mathrm{\infty }}{S}_n\mathrm{/}{n}^{2p-1}$ will be studied.