Moment Properties of the Multivariate Dirichlet Distributions

Let X1, …, Xn be real, symmetric, m×m random matrices; denote by Im the m×m identity matrix; and let a1, …, an be fixed real numbers such that aj>(m−1)/2, j=1, …, n. Motivated by the results of J. G. Mauldon (Ann. Math. Statist.30 (1959), 509–520) for the classical Dirichlet distributions, we consider the problem of characterizing the joint distribution of (X1, …, Xn) subject to the condition that E |Im−∑nj=1 TjXj|−(a1+…+an)=∏nj=1 |Im−Tj|−aj for all m×m symmetric matrices T1, …, Tn in a neighborhood of the m×m zero matrix. Assuming that the joint distribution of (X1, …, Xn) is orthogonally invariant, we deduce the following results: each Xj is positive-definite, almost surely; X1+…+Xn=Im, almost surely; the marginal distribution of the sum of any proper subset of X1, …, Xn is a multivariate beta distribution; and the joint distribution of the determinants (|X1|, …, |Xn|) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a1, …, an). In particular, for n=2 we obtain a new characterization of the multivariate beta distribution.