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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.

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