Characteristic property of a class of multivariate variance functions

Natural exponential families (NEFs) are well known to be characterized by their variance functions. A problem of increasing interest for dimension d > 1 is the following: given an open convex set Ω of (0,∞)d and a real analytic function V from Ω into the set of linear symmetric operators from ℝd, is V a variance function of some NEF? In the real line case of d = 1, this question was already solved. The aim of this work is to give necessary and sufficient conditions on V in order to be the variance function for some multivariate NEF. The notion of absolutely monotonic function on [0,∞)d is thus introduced, and the determination of moments of the NEF is also involved. For an NEF concentrated on [0,∞)d, a bridge is established between the behavior of V around of the origin and the existence conditions of the corresponding NEF. Some illustrating examples are presented.