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 - ∑j=1n TjXj−(a1+⋯+an) = ∏j=1n 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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty