Natural exponential families and generalized hypergeometric measures

Let be a positive Borel measure on n and pFq(a1, ap; b1, bq; s) be a generalized hypergeometric series. We define a generalized hypergeometric measure, p,q:=pFq(a1, ap; b1, bq;), as a series of convolution powers of the measure , and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968) family of distributions is an example of p,q in which is a Dirac measure on . For the case in which is a Dirac measure on n, we relate p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994). For pq, we show that certain measures p,q can be expressed as the convolution of a sequence of independent multi-dimensional Bernoulli trials. For p=q, q+1, we show that the measures p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.