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Functional limit theorems for Lévy processes satisfying Cramér’s condition. (English) Zbl 1244.60049
Summary: 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.

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