I.M. MacPhee
Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts
MacPhee, I.M.; Menshikov, M.V.; Wade, A.R.
Authors
Professor Mikhail Menshikov mikhail.menshikov@durham.ac.uk
Professor
Professor Andrew Wade andrew.wade@durham.ac.uk
Professor
Abstract
We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time τα from a wedge with apex at the origin and interior half-angle α by a non-homogeneous random walk on ℤ2 with mean drift at x of magnitude O(∥x∥−1) as ∥x∥→∞. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors stated that τα<∞ a.s. for any α. Here we study the more difficult problem of the existence and non-existence of moments , s>0. Assuming a uniform bound on the walk’s increments, we show that for α<π/2 there exists s0∈(0,∞) such that TeX is finite for ss0; under specific assumptions on the drift field, we show that we can attain TeX for any s>1/2. We show that there is a phase transition between drifts of magnitude O(∥x∥−1) (the critical regime) and o(∥x∥−1) (the subcritical regime). In the subcritical regime, we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.
Citation
MacPhee, I., Menshikov, M., & Wade, A. (2013). Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts. Journal of Theoretical Probability, 26(1), 1-30. https://doi.org/10.1007/s10959-012-0411-x
Journal Article Type | Article |
---|---|
Publication Date | Mar 1, 2013 |
Deposit Date | Feb 6, 2013 |
Publicly Available Date | Feb 14, 2013 |
Journal | Journal of Theoretical Probability |
Print ISSN | 0894-9840 |
Electronic ISSN | 1572-9230 |
Publisher | Springer |
Peer Reviewed | Peer Reviewed |
Volume | 26 |
Issue | 1 |
Pages | 1-30 |
DOI | https://doi.org/10.1007/s10959-012-0411-x |
Public URL | https://durham-repository.worktribe.com/output/1468656 |
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Copyright Statement
The original publication is available at www.springerlink.com
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