Characterization of the Dirichlet distribution on symmetric matrices

A remarkable characterization result concerning the real-Dirichlet distribution says that if X1,...,Xq are real random variables, then (X1,...,Xq) has a Dirichlet joint distribution with parameters (p1,...,pq) if and only if, for all positive real numbers f1,...,fq,The aim of the present paper is to extend this characterization to the Dirichlet distributions on positive definite symmetric matrices defined in Letac et al. [2001. An expectation formula for the multivariate Dirichlet distribution. J. Multivariate Anal. 77, 117-137].