Recurrence and transience of multidimensional elephant random walks

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on ${\mathbb{Z}}^d$($d\ge 3$) 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 $p=(2d+1)/(4d)$. Let $S$ be an elephant random walk with parameter $p$. For $p\le 3/4$, 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.

arXiv comment36 pages, 2 figures

References in corpus (13)

• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• Elephant Random Walks and their connection to Pólya-type urns
• A martingale approach for the elephant random walk
• Central Limit Theorem for the Elephant Random Walk
• Random Recursive Trees and the Elephant Random Walk
• A strong invariance principle for the elephant random walk
• Gaussian fluctuation for superdiffusive elephant random walks
• Limit distributions for large Pólya urns
• The Shark Random Swim (Lévy flight with memory)
• The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes
• Joint invariance principles for random walks with positively and negatively reinforced steps
• Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks
• Recurrence of the plane Elephant random walk