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Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

MacPhee, I.M. and Menshikov, Mikhail V. and Wade, A.R. (2010) Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov Processes and Related Fields, 16 (2). pp. 351-388. ISSN 1024-2953

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Abstract

We study the rst exit time from an arbitrary cone with apex at the origin by a non-homogeneous 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&#x100000;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&#x100000;, 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.