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Les familles exponentielles à variance quadratique homogène sont des lois de Wishart sur un cône symétrique. (Exponential families with a homogeneous quadratic variance are Wishart distributions on symmetric cones). (French) Zbl 0745.62051
Summary: The aim of this note is to characterize the exponential families on \(\mathbb{R}^ n\) with a variance-function which is quadratic and homogeneous in the mean, i.e. \(V(m_ 1, \dots ,m_ n)= \sum_{i,j=1}^ n A_{ij} m_ i m_ j\), where \(A_{ij}\) are real \((n,n)\) symmetric matrices. They are defined on symmetric cones and thus are products of Wishart families on the five irreducible symmetric cones.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions
60B11 Probability theory on linear topological spaces