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

MacPhee, Iain M.; Menshikov, Mikhail V.; Wade, Andrew R.

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.

Citation

MacPhee, I. M., Menshikov, M. V., & Wade, A. R. (2010). Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov processes and related fields, 16(2), 351-388

• Journal Article Type: Article
• Publication Date: Jan 1, 2010
• Deposit Date: Oct 17, 2012
• Publicly Available Date: Feb 20, 2013
• Journal: Markov processes and related fields.
• Print ISSN: 1024-2953
• Publisher: Polymat
• Peer Reviewed: Peer Reviewed
• Volume: 16
• Issue: 2
• Pages: 351-388
• Public URL: https://durham-repository.worktribe.com/output/1472696
• Publisher URL: http://mech.math.msu.su/~malyshev/abs10.htm