Cutoff Phenomenon for Nearest Lamperti’s Random Walk

Abstract

We consider transient neighbor random walks on the positive part of the real line, the transition probability is state dependent being a special case of the Lamperti’s random walk. We show that a sequence of lazy random walks on [0, n] exhibits cutoff phenomenon. As an important step in the proof, we derive the limit speed of the expectation and variance of the hitting times of the random walk exactly. And as a byproduct, we give a probabilistic proof for the law of large numbers of the random walk which has been obtained by Voit (1992) using the method of polynomial hypergroups.