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

I.M. MacPhee, Mikhail V. Menshikov, Andrew Wade

Research output: Contribution to journalArticle

3 Citations (Scopus)

### 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.
Language English 1-30 30 Journal of Theoretical Probability 26 1 10.1007/s10959-012-0411-x Published - Mar 2013

### Fingerprint

Exit Time
Wedge
Random walk
Moment
Zero
First Exit Time
Uniform Bound
Apex
Walk
Increment
Brownian motion
Interior
Phase Transition
Asymptotic Behavior
Regularity
Exit
Analogue
Angle
Theorem

### Keywords

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

### Cite this

MacPhee, I.M. ; Menshikov, Mikhail V. ; Wade, Andrew. / Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts. In: Journal of Theoretical Probability. 2013 ; Vol. 26, No. 1. pp. 1-30.
title = "Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts",
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.",
keywords = "angular asymptotics, non-homogeneous random walks, passage-time moments , lyapunov functions",
author = "I.M. MacPhee and Menshikov, {Mikhail V.} and Andrew Wade",
year = "2013",
month = "3",
doi = "10.1007/s10959-012-0411-x",
language = "English",
volume = "26",
pages = "1--30",
journal = "Journal of Theoretical Probability",
issn = "0894-9840",
number = "1",

}

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

In: Journal of Theoretical Probability, Vol. 26, No. 1, 03.2013, p. 1-30.

Research output: Contribution to journalArticle

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.

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 -