Characterization of the Dirichlet distribution on symmetric matrices

Abstract

A remarkable characterization result concerning the real-Dirichlet distribution says that if X 1 , … , X q are real random variables, then ( X 1 , … , X q ) has a Dirichlet joint distribution with parameters ( p 1 , … , p q ) if and only if, for all positive real numbers f 1 , … , f q , (0.1) E ∑ i = 1 q f i X i - ( p 1 + ⋯ + p q ) = ∏ i = 1 q f i - p i . 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].