Recurrence and transience of multidimensional elephant random walks

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on Zᵈ(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 Sₙ. For $p>3/4$, the distribution of limn→ ∞ Sₙ/n²ᵖ⁻¹ will be studied.