Excursions and local limit theorems for Bessel-like random walks
Probability
2015-03-13 v2
Abstract
We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -\delta/2x + o(1/x) with \delta > -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}\phi(n), where c = (3+\delta)/2 and \phi is a slowly varying function given explicitly in terms of the o(1/x) terms.
Cite
@article{arxiv.0912.4550,
title = {Excursions and local limit theorems for Bessel-like random walks},
author = {Kenneth S. Alexander},
journal= {arXiv preprint arXiv:0912.4550},
year = {2015}
}
Comments
44 pages. Numerous small corrections and clarifications. References added