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

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

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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.
Original languageEnglish
Pages (from-to)351-388
Number of pages38
JournalMarkov Processes and Related Fields
Volume16
Issue number2
Publication statusPublished - Jul 2010

Keywords

  • asymptotic direction
  • exit from cones
  • inhomogeneous random walk
  • perturbed random walk
  • random walk in random environment

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