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

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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 
Keywords:  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:  16 Apr 2015 09:29 
Related URLs:  
URI:  http://strathprints.strath.ac.uk/id/eprint/27284 
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