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.
"Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition." Electron. J. Probab. 16 2020 - 2038, 2011. https://doi.org/10.1214/EJP.v16-930