The class of Tweedie exponential dispersion models includes such well-known continuous distributions as the normal and gamma, the purely discrete sclaled Poisson distribution, as well as the class of mixed compound Poisson-Gamma distributions which have positive mass at zero, but are otherwise continuous. The remaining Tweedie models are derived by exponential tilting of extreme stable distributions. This quite heterogeneous class of distributions was introduced in statistics by Tweedie (1984),mainly because of the simple form of their unit variance function: V(m) = m^p.

It is known (see, e.g., Jorgensen (1997)) that Tweedie models possess scaling properties similar to those of the stable laws and also that they emerge as weak limits of appropriately scaled natural exponential families. Domains of attraction to Tweedie models are described in Jorgensen, Martinez and Tsao (1994) and Jorgensen, Martinez, Vinogradov (1998). At the same time, Jorgensen (1997, pp. 150-151) and Wentzell (1998,personal communication) independently conjectured that the classical theorems on weak convergence to stable laws and those on weak convergence to Tweedie models should be related.

In this lecture, it will be shown how the theorems on weak convergence to Tweedie models with index p>2 can be derived from those on weak convergence to the positive stable laws.