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

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

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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.
Original languageEnglish
Pages (from-to)1-30
Number of pages30
JournalJournal of Theoretical Probability
Volume26
Issue number1
DOIs
Publication statusPublished - Mar 2013

Keywords

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

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