Electronic Journal of Probability

Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition

Matyas Barczy and Jean Bertoin

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We consider a Lévy process that starts from $x<0$ and conditioned on having a positive maximum. When Cramér's condition holds, we provide two weak limit theorems as $x$ goes to $-\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 73, 2020-2038.

Accepted: 31 October 2011
First available in Project Euclid: 1 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G18: Self-similar processes 60B10: Convergence of probability measures

Lévy process Cramér's condition self-similar Markov process

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Barczy, Matyas; Bertoin, Jean. Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition. Electron. J. Probab. 16 (2011), paper no. 73, 2020--2038. doi:10.1214/EJP.v16-930. https://projecteuclid.org/euclid.ejp/1464820242

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