Angular asymptotics for multidimensional nonhomogeneous random walks with asymptotically zero drift
MacPhee, I.M. and Menshikov, Mikhail V. and Wade, A.R. (2010) Angular asymptotics for multidimensional nonhomogeneous random walks with asymptotically zero drift. Markov Processes and Related Fields, 16 (2). pp. 351388. ISSN 10242953 (http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.1772...)
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Abstract
We study the rst exit time from an arbitrary cone with apex at the origin by a nonhomogeneous random walk (Markov chain) on Zd (d 2) with mean drift that is asymptotically zero. Specically, if the mean drift at x 2 Zd is of magnitude O(kxk􀀀1), we show that < 1 a.s. for any cone. On the other hand, for an appropriate drift eld with mean drifts of magnitude kxk􀀀, 2 (0; 1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.


Item type: Article ID code: 27284 Dates: DateEventJuly 2010PublishedKeywords: asymptotic direction, exit from cones, inhomogeneous random walk, perturbed random walk, random walk in random environment, Probabilities. Mathematical statistics, Statistics and Probability, Applied Mathematics Subjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Mrs Carolynne Westwood Date deposited: 31 Aug 2010 12:55 Last modified: 01 Nov 2023 09:40 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/27284