Excursions and Local Limit Theorems for Bessel-like Random Walks

Kenneth S. Alexander (University of Southern California)


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 at a given time (uniformly in a large range of heights.) In particular, for certain drifts inversely proportional to $x$ up to smaller-order correction terms, we show that the probability of a first return to $0$ at time $n$ decays as a certain inverse power of $n$, multiplied by a slowly varying factor that depends on the drift correction terms.

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Pages: 1-44

Publication Date: January 2, 2011

DOI: 10.1214/EJP.v16-848


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