Stability of the Overshoot for Levy Processes

We give equivalences for conditions like X(T(r))/r → 1 and X(T*(r))/ r → 1, where the convergence is in probability or almost sure, both as r → 0 and r → ∞, where X is a Levy 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,...

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Bibliographic Details
Published in:The Annals of probability 2002-01, Vol.30 (1), p.188-212
Main Authors: Doney, R. A., Maller, R. A.
Format: Article
Language:English
Subjects:
60G17
60G51
Exact sciences and technology
exit times
first passage times
local behavior
Mathematics
Probability and statistics
Probability theory and stochastic processes
Processes with independent increments
Random walk
Sciences and techniques of general use
Stochastic processes
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Summary:We give equivalences for conditions like X(T(r))/r → 1 and X(T*(r))/ r → 1, where the convergence is in probability or almost sure, both as r → 0 and r → ∞, where X is a Levy 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 → 1 a.s. as r → 0 is equivalent to X "creeping" across a level.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1020107765