A Classification of Reproducible Natural Exponential Families in the Broad Sense

Let μ be a positive Radon measure on ℝ having a Laplace transform L_μ and F = F(μ) be a natural exponential family (NEF) generated by μ. A positive Radon measure μ_λ with λ > 0 and its associated NEF F_λ = F(μ _λ) are called the λth convolution powers of μ and F, respectively, if L_μλ = L_μ ^λ. Let f_α,β(F) be the image of an NEF F under the affine transformation f_α,β: x → αx + β. If for a given NEF F, there exists a triple (α, β, λ) in ℝ³ such that f_α,β(F) = F _λ, we call F a reproducible NEF in the broad sense. In other words, an NEF F is reproducible in the broad sense if a convolution power of F equals an affine transformation of F. Clearly, for λ = 1, F is reproducible in the broad sense if there exists an affinity under which F is invariant. In this paper we obtain a complete classification of the class of reproducible NEF's in the broad sense. We show that this class is composed of infinitely divisible NEF's and that it contains the class of NEF's having exponential and power variance functions as well as NEF's constituting discrete versions of the latter NEF's. We also provide a characterization of the reproducible NEF's in the broad sense in terms of their associated exponential dispersion models.