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

MacPhee, Iain M. and Menshikov, Mikhail V. and Wade, Andrew 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.

## Abstract

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-\beta}$, $\beta \in (0,1)$, 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 $2$nd 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 (VoR) Version of Record Download PDF (387Kb) Peer-reviewed http://mech.math.msu.su/~malyshev/abs10.htm No date available 20 February 2013 2010 No date available