### Abstract

We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\tau_\alpha$ from a wedge with apex at the origin and interior half-angle $\alpha$ by a non-homogeneous random walk on $\Z^2$ with mean drift at $\bx$ of magnitude $O( \| \bx \|^{-1})$ as $\| \bx \| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors stated that $\tau_\alpha < \infty$ a.s.\ for any $\alpha$. Here we study the more difficult problem of the existence and non-existence of moments $\Exp [ \tau_\alpha^s]$, $s>0$. Assuming a uniform bound on the walk's increments, we show that for $\alpha < \pi/2$ there exists $s_0 \in (0,\infty)$ such that $\Exp [ \tau_\alpha^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $\Exp [ \tau_\alpha ^s] = \infty$ for any $s > 1/2$.

We show that there is a phase transition between drifts of magnitude $O(\| \bx \|^{-1})$ (the critical regime) and $o( \| \bx \|^{-1} )$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.

We show that there is a phase transition between drifts of magnitude $O(\| \bx \|^{-1})$ (the critical regime) and $o( \| \bx \|^{-1} )$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.

Original language | English |
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Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Journal of Theoretical Probability |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2013 |

### Keywords

- angular asymptotics
- non-homogeneous random walks
- passage-time moments
- lyapunov functions

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## Cite this

MacPhee, I. M., Menshikov, M. V., & Wade, A. (2013). Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts.

*Journal of Theoretical Probability*,*26*(1), 1-30. https://doi.org/10.1007/s10959-012-0411-x