NATURAL EXPONENTIAL-FAMILIES WITH AN HOMOGENEOUS QUADRATIC VARIANCE ARE WISHART DISTRIBUTIONS ON SYMMETRICAL CONES

The aim of this Note is to characterize the exponential families on R(n) with a variance-function which is quadratic and homogeneous in the mean, i.e. V(m1, ..., Vm1), ....,m(n)) = SIGMA A(ij)m(i)m(j) where A(ij) are real (n, n) symmetric matrices. They are all defined on symmetric cones and thus are products of Wishart families on the five irreducible symmetric cones.