Moments of exit times from wedges for nonhomogeneous random walks with asymptotically zero drifts
MacPhee, I.M. and Menshikov, Mikhail V. and Wade, Andrew (2013) Moments of exit times from wedges for nonhomogeneous random walks with asymptotically zero drifts. Journal of Theoretical Probability, 26 (1). pp. 130. ISSN 08949840 (https://doi.org/10.1007/s109590120411x)
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Abstract
We study quantitative asymptotics of planar random walks that are spatially nonhomogeneous 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 halfangle $\alpha$ by a nonhomogeneous 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 nonexistence 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 nonhomogeneous 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.


Item type: Article ID code: 43390 Dates: DateEventMarch 2013PublishedSubjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 02 Apr 2013 15:37 Last modified: 08 Apr 2024 19:50 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/43390