Excursions and local limit theorems for Bessel-like random walks

2009/12/23 by Kenneth S. Alexander, Alexander, Kenneth S.
Economics, Econometrics and Finance · Mathematics · #60J10 #60J80 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60J10 #msc:60J80

paper · pdf · doi:10.48550/arxiv.0912.4550

44 pages. Numerous small corrections and clarifications. References added

openalex publication_date 2009/12/23 · arxiv created 2010/09/03 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 -δ/2x + o(1/x) with δ> -1, we show that the probability of a first return to 0 at time n is asymptotically n^-cϕ(n), where c = (3+δ)/2 and ϕis a slowly varying function given explicitly in terms of the o(1/x) terms.

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