## The Annals of Probability

### Stability of the Overshoot for Lévy Processes

#### 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.

#### Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 188-212.

Dates
First available in Project Euclid: 29 April 2002

https://projecteuclid.org/euclid.aop/1020107765

Digital Object Identifier
doi:10.1214/aop/1020107765

Mathematical Reviews number (MathSciNet)
MR1894105

Zentralblatt MATH identifier
1016.60052

#### Citation

Doney, R.A.; Maller, R.A. Stability of the Overshoot for Lévy Processes. Ann. Probab. 30 (2002), no. 1, 188--212. doi:10.1214/aop/1020107765. https://projecteuclid.org/euclid.aop/1020107765

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