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.
LanguageEnglish
Pages1-30
Number of pages30
JournalJournal of Theoretical Probability
Volume26
Issue number1
DOIs
Publication statusPublished - 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.
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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

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AU - Menshikov, Mikhail V.

AU - Wade, Andrew

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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.

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