%0 Journal Article
%T Excursions and local limit theorems for Bessel-like random walks
%A Kenneth S. Alexander
%J Mathematics
%D 2009
%I arXiv
%X 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.
%U http://arxiv.org/abs/0912.4550v2