Moment properties of the multivariate Dirichlet distributions

Let X-1,...,X-n, be real, symmetric, m x m random matrices; denote by I,, the m x m identity matrix; and let a,..., a,, 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 (X-1,...,X-n) subject to the condition that E\I-m - Sigma(j=1)(n)T(j)X(j)\(-(a1+...+an)) = Pi(j=1)(n)\I-m - T-j\(-aj) for all m x m symmetric matrices T-1,...,T-n in a neighborhood of the rn x m zero matrix. Assuming that the joint distribution of (X-1,...,X-n) is orthogonally invariant, we deduce the following results: each X-j is positive-definite, almost surely; X-1+...+X-n = I-m almost surely; the marginal distribution of the sum of any proper subset of X,..., X,, is a multivariate beta distribution; and the joint distribution of the determinants (\X-1\,....,\X-n\) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a(1),...,a(n)). In particular, for n = 2 we obtain a new characterization of the multivariate beta distribution. (C) 2002 Elsevier Science (USA).