### Abstract

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.

Language | English |
---|---|

Pages | 1-30 |

Number of pages | 30 |

Journal | Journal of Theoretical Probability |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2013 |

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### Keywords

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

### Cite this

*Journal of Theoretical Probability*,

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

}

*Journal of Theoretical Probability*, vol. 26, no. 1, pp. 1-30. https://doi.org/10.1007/s10959-012-0411-x

**Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts.** / MacPhee, I.M.; Menshikov, Mikhail V.; Wade, Andrew.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

AU - MacPhee, I.M.

AU - Menshikov, Mikhail V.

AU - Wade, Andrew

PY - 2013/3

Y1 - 2013/3

N2 - 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.

AB - 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.

KW - angular asymptotics

KW - non-homogeneous random walks

KW - passage-time moments

KW - lyapunov functions

UR - http://www.scopus.com/inward/record.url?scp=84874526443&partnerID=8YFLogxK

U2 - 10.1007/s10959-012-0411-x

DO - 10.1007/s10959-012-0411-x

M3 - Article

VL - 26

SP - 1

EP - 30

JO - Journal of Theoretical Probability

T2 - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 1

ER -