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

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

Markov Processes and Related Fields, 16, no. 2, July 2010, 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 $\mathbb{Z}^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\mathbf{x} \in \mathbb{Z}^d$ is of magnitude $O(\| \mathbf{x}\|^{-1})$, we show that $\tau<\infty$ a.s.~for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \mathbf{x} \|^{-\beta}$, $\beta \in (0,1)$, 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.