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Excursions and path functionals for stochastic processes with asymptotically zero drifts

Hryniv, Ostap; Menshikov, Mikhail V.; Wade, Andrew R.

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Abstract

We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form , α>0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional ‘centrally biased’ random walks on Rd; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.

Journal Article Type Article
Publication Date Jun 1, 2013
Deposit Date Feb 12, 2013
Publicly Available Date Feb 27, 2013
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 123
Issue 6
Pages 1891-1921
DOI https://doi.org/10.1016/j.spa.2013.02.001
Keywords Path functional, Excursion, Maximum, Passage-time, Additive functional, Path integral, Lamperti’s problem, Centre of mass, Centrally biased random walk.
Public URL https://durham-repository.worktribe.com/output/1489295

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Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic processes and their applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic processes and their applications, 123, 6, 2013, 10.1016/j.spa.2013.02.001






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