Excursions and local limit theorems for Bessel-like random walks

TL;DR

This paper derives explicit asymptotic probabilities for Bessel-like reflecting random walks on nonnegative integers, including hitting times and position distributions, revealing detailed local limit behaviors influenced by drift.

Contribution

It provides new explicit asymptotic formulas for hitting times and position probabilities of Bessel-like random walks with specific drift conditions.

Findings

Asymptotic probability of first return to 0 at time n is n^{-(3+ extdelta)/2} times a slowly varying function.

Explicit formulas for the distribution of the walk's position at time n.

Characterization of the effect of drift on local limit behaviors.

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.