Doney, R. A. and Maller, R.A.
(2002)
*Stability of the overshoot for Lévy processes.*
The Annals of Probability, 30 (1).
pp. 188-212.
ISSN 0091-1798

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## Abstract

We give equivalences for conditions like $X(T(r))/r\rightarrow 1$ and $X(T^{*}(r))/\allowbreak r\rightarrow 1$, where the convergence is in probability or almost sure, both as $r\rightarrow 0$ and $r\rightarrow \infty$, where $X$ is a L\'{e}vy process and $T(r)$ and $T^{*}(r)$ are the first exit times of $X$ out of the strip $\{(t,y):t> 0,|y|\leq r\}$ and half-plane $\{(t,y):t> 0$, $y\leq r\}$, respectively. We also show, using a result of Kesten, that $X(T^{*}(r))/r\rightarrow 1$ a.s.\ as $r\to 0$ is equivalent to $X$ ``creeping'' across a level.

Item Type: | Article |
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Uncontrolled Keywords: | Processes with independent increments; exit times; first passage times; local behavior |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes |

Depositing User: | Ms Lucy van Russelt |

Date Deposited: | 06 Sep 2006 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/588 |

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