Generalized Variance Functions for Infinitely Divisible Mixture Distributions

This paper deals with the characterization of a class of infinitely divisible mixture distributions when the mixing parameter is the power of convolution. In framework of natural exponential families, we give the expression of its variance function. Furthermore, we explicit its generalized variance function which is the determinant of the covariance matrix and, then, we determine its associated L,vy measure. Some important examples of multivariate mixture of discrete distributions are given. Our examples introduce an infinitely divisible family of multivariate discrete models that are lacking in the literature.