Excursions and local limit theorems for Bessel-like random walks - Mathematics > Probability

Abstract: We consider reflecting random walks on the nonnegative integers with drift oforder 1-x at height x. We establish explicit asymptotics for variousprobabilities associated to such walks, including the distribution of thehitting time of 0 and first return time to 0, and the probability of being at agiven height k at time n uniformly in a large range of k. In particular, fordrift of form -\delta-2x + o1-x with \delta > -1, we show that theprobability of a first return to 0 at time n is asymptotically n^{-c}\phin,where c = 3+\delta-2 and \phi is a slowly varying function given explicitlyin terms of the o1-x terms.

Author: Kenneth S. Alexander

Source: https://arxiv.org/