ABSTRACT
The class of Tweedie exponential dispersion models generalizes positive and extreme stable distributions. They were used in the statistical analysis of car insurance claims by Jørgensen and Souza (1994), and a new type of weak convergence theorem to the Tweedie models was proved by Jørgensen, Martínez and Tsao (1994). We discuss some analogies between the development of this theory and certain phase-transition phenomena in the theory of large deviations, branching processes, mathematical physics and actuarial mathematics. Thus, we characterize a new part of the domains of attraction to the Tweedie families, and suggest certain probability models for analyzing some insurance claims data. Specifically, distributions with exponential-power tails belong to the domains of attraction to certain Tweedie models, and the same class of distributions reveals some critical-point properties.
