A class of infinitely divisible variance functions with an application to the polynomial case

Let F be a natural exponential family on ??? with variance function (V, Ω). Here, Ω is the mean domain of F and V is its variance expressed in terms of the mean μ ε{lunate} Ω. In this note we prove the following result. Consider an open interval Ω = (0, b), 0 < b ≤ ∞, and a positive real analytic function V on Ω. If V² is absolutely monotone on [0, b) and V has the form μ^αt(μ), where α ≥ 1 and t is real analytic in a neighborhood of zero, then there exits an infinitely divisible natural exponential family with variance function (V, Ω). We illustrate this result with several examples of general nature.