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 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.
Citation
Kenneth Alexander. "Excursions and Local Limit Theorems for Bessel-like Random Walks." Electron. J. Probab. 16 1 - 44, 2011. https://doi.org/10.1214/EJP.v16-848
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