Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition

Matyas Barczy (University of Debrecen)
Jean Bertoin (Université Pierre et Marie Curie)


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.

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Pages: 2020-2038

Publication Date: October 31, 2011

DOI: 10.1214/EJP.v16-930


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