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The inversion of a Levy measure was first introduced (under a different name)in Sato 2007. We generalize the definition and give some properties. We thenuse inversions to derive a relationship between weak convergence of a Levyprocess to an infinite variance stable distribution when time approaches zeroand weak convergence of a different Levy process as time approaches infinity.This allows us to get self contained conditions for a Levy process to convergeto an infinite variance stable distribution as time approaches zero. Weformulate our results both for general Levy processes and for the importantclass of tempered stable Levy processes. For this latter class, we givedetailed results in terms of their Rosinski measures.

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