Natural Exponential Families and Generalized Hypergeometric Measures

Let ν be a positive Borel measure on R and pFq(a1, . . . , ap; b1, . . . , bq; s) be a generalized hypergeometric series. We define a generalized hypergeometric measure, μp,q := pFq(a1, . . . , ap; b1, . . . , bq; ν), as a series of convolution powers of the measure ν, and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp family of distributions (Sankhyā, Ser. A, 30, (1968), 401–410) is an example of μp,q in which ν is a Dirac measure on R. For the case in which ν is a Dirac measure on R, we relate μp,q to the diagonal natural exponential families classified by Bar-Lev, et al. (J. Theoret. Probab. 7 (1994), 883-929). For p < q we show that certain measures μp,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 μp,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures. AMS 2000 Subject Classification: 60E05, 62E10, 62E17, 62D05.