### Abstract

Let X_{1}, ⋯, X_{n} be real, symmetric, m × m random matrices; denote by I_{m} the m × m identity matrix; and let a_{1}, ⋯, 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 (X_{1}, ⋯, X_{n}) subject to the condition that E I_{m} - ∑_{j=1}^{n} T_{j}X_{j}^{−(a1+⋯+an)} = ∏_{j=1}^{n} I_{m} - T_{j}^{−aj} for all m × m symmetric matrices T_{1}, ⋯, T_{n} in a neighborhood of the m × 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_{1}, ⋯, X_{n} 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.

Original language | English (US) |
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Pages (from-to) | 240-262 |

Number of pages | 23 |

Journal | Journal of Multivariate Analysis |

Volume | 82 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2002 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Multivariate Analysis*,

*82*(1), 240-262. https://doi.org/10.1006/jmva.2001.2016