Iain M. MacPhee
Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift
MacPhee, Iain M.; Menshikov, Mikhail V.; Wade, Andrew R.
Authors
Professor Mikhail Menshikov mikhail.menshikov@durham.ac.uk
Professor
Professor Andrew Wade andrew.wade@durham.ac.uk
Professor
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 |
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