Excursions and local limit theorems for Bessel-like random walks

Details Excursions and local limit theorems for Bessel-like random walks Excursions and local limit theorems for Bessel-like random walks

2009-12-23

arxiv.org/abs/0912.4550v2

Mathematics

Probability

44 pages. Numerous small corrections and clarifications. References added

Scientific paper

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.

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