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Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

MacPhee, I.M.; Menshikov, M.V.; Wade, A.R.

Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts Thumbnail


Authors

I.M. MacPhee



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.

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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The original publication is available at www.springerlink.com





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