TAUBERIAN RESULTS FOR DENSITIES WITH GAUSSIAN TAILS

We study a class of probability densities with very thin upper tails. These densities generate exponential families which are asymptotically normal. Furthermore the class is closed under convolution. In this pa per we shall be concerned with Abelian and strong Tauberian theorems f or moment generating functions and Laplace transforms with respect to these densities. We obtain a duality relation between this class of de nsities and the associated class of moment generating functions which is closely related to the duality relation for convex functions.