On Infinitely Divisible Exponential Dispersion Model Related to Poisson-Exponential Distribution

Abstract

We construct a univariate exponential dispersion model comprised of discrete infinitely divisible distributions. This model emerges in the theory of branching processes. We obtain a representation for the Lévy measure of relevant distributions and characterize their laws as Poisson mixtures and/or compound Poisson distributions. The regularity of the unit variance function of this model is employed for the derivation of approximations by the Poisson-exponential model. We emphasize the role of the latter class. We construct local approximations relating them to properties of special functions and branching diffusions.

Keywords:

• Bessel function
• Branching process
• Compound Poisson distribution
• Confluent hypergeometric function
• Lévy measure
• Local approximation
• Poisson mixture
• Pólya-Aeppli distribution
• Power-variance family
• Rao damage process
• Tweedie exponential dispersion models
• Unit variance function
• Weak convergence

Mathematics Subject Classification:

• Primary 60E07, 60F05
• Secondary 60J80, 62E20

Acknowledgments

I thank D. Dawson, A. Feuerverger, L. Mytnik, and T. Salisbury for help and the anonymous referee for many useful suggestions. I appreciate the hospitality of the Fields Institute, the University of Toronto, and York University.