Tauber theory for infinitely divisible variance functions

Abstract

We study a notion of Tauber theory for infinitely divisible natural exponential families, showing that the variance function of the family is (bounded) regularly varying if and only if the canonical measure of the Lévy-Khinchine representation of the family is (bounded) regularly varying. Here a variance function V is called bounded regularly varying if V(μ)\sim cμ^p either at zero or infinity, with a similar definition for measures. The main tool of the proof is classical Tauber theory.