### 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 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Multivariate Analysis*,

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

}

*Journal of Multivariate Analysis*, vol. 82, no. 1, pp. 240-262. https://doi.org/10.1006/jmva.2001.2016

**Moment properties of the multivariate Dirichlet distributions.** / Gupta, Rameshwar D.; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Moment properties of the multivariate Dirichlet distributions

AU - Gupta, Rameshwar D.

AU - Richards, Donald

PY - 2002/1/1

Y1 - 2002/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0036334840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036334840&partnerID=8YFLogxK

U2 - 10.1006/jmva.2001.2016

DO - 10.1006/jmva.2001.2016

M3 - Article

AN - SCOPUS:0036334840

VL - 82

SP - 240

EP - 262

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -