Inversions of Lévy Measures and the Relation Between Long and Short Time Behavior of Lévy Processes

The inversion of a Lévy measure was first introduced (under a different name) in Sato (ALEA Lat Am J Probab Math Stat 3:67–110, 2007). We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Lévy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Lévy process as time approaches infinity. This allows us to get self-contained conditions for a Lévy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Lévy processes and for the important class of tempered stable Lévy processes. For this latter class, we give detailed results in terms of their Rosiński measures.