Functional limit theorems for Lévy processes satisfying Cramér's condition

Abstract

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 −∞ for the law of the (two-sided) path shifted at the first instant when it enters (0,∞), 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.