Excursions and path functionals for stochastic processes with asymptotically zero drifts

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

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.

Citation

Hryniv, O., Menshikov, M. V., & Wade, A. R. (2013). Excursions and path functionals for stochastic processes with asymptotically zero drifts. Stochastic Processes and their Applications, 123(6), 1891-1921. https://doi.org/10.1016/j.spa.2013.02.001

• 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
• Electronic ISSN: 1879-209X
• 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