### 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􀀀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􀀀, 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.

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􀀀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􀀀, 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 language | English |
---|---|

Pages (from-to) | 351-388 |

Number of pages | 38 |

Journal | Markov Processes and Related Fields |

Volume | 16 |

Issue number | 2 |

Publication status | Published - Jul 2010 |

### Keywords

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

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## Cite this

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.