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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.

##### MSC:
 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