Dispersion models for geometric sums

Abstract

A new class of geometric dispersion models associated with geometric sums is introduced by combining a geometric tilting operation with geometric compounding, in much the same way that exponential dispersion models combine exponential tilting and convolution. The construction is based on a geometric cumulant function which characterizes the geometric compounding operation additively. The so-called v-function is shown to be a useful characterization and convergence tool for geometric dispersion models, similar to the variance function for natural exponential families. A new proof of Rényi’s theorem on convergence of geometric sums to the exponential distribution is obtained, based on convergence of v-functions. It is shown that power v-functions correspond to a class of geometric Tweedie models that appear as limiting distributions in a convergence theorem for geometric dispersion models with power asymptotic v-functions. Geometric Tweedie models include geometric tiltings of Laplace, Mittag-Leffler and geometric extreme stable distributions, along with geometric versions of the gamma, Poisson and gamma compound Poisson distributions.