Functional limit theorems for Lévy processes satisfying Cramér's condition

Matyas Barczy (1) , Jean Bertoin (2)

1 Department of Applied Mathematics and Probability
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $x\to -\infty$ 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.

Keywords: Lévy process; Cramér's condition; self-similar Markov process

Domains: Probability [math.PR]

Matyas Barczy, Jean Bertoin. Functional limit theorems for Lévy processes satisfying Cramér's condition. 2011. ⟨hal-00588557⟩