Domains of attraction to Tweedie distributions

We derive new Tauberian theorems for natural exponential families, characterizing regularity properties of the family's variance function in terms of those of an extreme generating measure. This provides normal and general domains of attraction to Tweedie distributions and leads to new results on weak convergence of natural exponential families to Tweedie distributions, parallel to weak convergence toward stable laws. In particular, we give the domains of attraction to the gamma and compound Poisson-gamma distributions.