Moment Properties of the Multivariate Dirichlet Distributions

Let X1, ..., Xn be real, symmetric, mxm random matrices; denote by Im the mxm 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  Im-[summation operator]nj=1 TjXj-(a1+...+an)=[product operator]nj=1 Im-Tj-aj for all mxm symmetric matrices T1, ..., Tn in a neighborhood of the mxm 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.