Moment properties of the multivariate Dirichlet distributions

Let X₁, ⋯, X_n be real, symmetric, m × m random matrices; denote by I_m the m × m identity matrix; and let a₁, ⋯, a_n be fixed real numbers such that a_j > (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, ⋯, X_n) subject to the condition that E I_m - ∑_j=1ⁿ T_jX_j^{−(a1+⋯+an)} = ∏_j=1ⁿ I_m - T_j^−aj for all m × m symmetric matrices T₁, ⋯, T_n in a neighborhood of the m × m zero matrix. Assuming that the joint distribution of (X₁, ⋯, X_n) is orthogonally invariant, we deduce the following results: each X_j is positive-definite, almost surely; X₁ +⋯ + X_n = I_m, almost surely; the marginal distribution of the sum of any proper subset of X₁, ⋯, X_n is a multivariate beta distribution; and the joint distribution of the determinants (X₁, ⋯, X_n) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a₁, ⋯, a_n). In particular, for n = 2 we obtain a new characterization of the multivariate beta distribution.