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Excursions and Local Limit Theorems for Bessel-like Random Walks

   
@article{EJP848,
	author = {Kenneth Alexander},
	title = {Excursions and Local Limit Theorems for Bessel-like Random Walks},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {16},
	year = {2011},
	keywords = {excursion, Lamperti problem, random walk, Bessel process},
	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.},
	pages = {no. 1, 1-44},
	issn = {1083-6489},
	doi = {10.1214/EJP.v16-848},    
        url = {http://ejp.ejpecp.org/article/view/848}}