The Annals of Statistics

The complex Wishart distribution and the symmetric group

Piotr Graczyk, Gérard Letac, and Hélène Massam

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Abstract

Let V be the space of (r,r) Hermitian matrices and let $\Omega$ be the cone of the positive definite ones. We say that the random variable S, taking its values in $\overline{\Omega},$ has the complex Wishart distribution $\gamma_{p,\sigma}$ if $\mathbb{E}(\exp \,\tr (\theta S))=(\det (I_r-\sigma\theta))^{-p},$ where $\sigma$ and $\sigma^{-1}-\theta$ are in $\Omega,$ and where p=1,2,...,r-1 or p>r-1. In this paper, we compute all moments of $S$ and $S^{-1}.$ The techniques involve in particular the use of the irreducible characters of the symmetric group.

Article information

Source
Ann. Statist., Volume 31, Number 1 (2003), 287-309.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1046294466

Digital Object Identifier
doi:10.1214/aos/1046294466

Mathematical Reviews number (MathSciNet)
MR1962508

Zentralblatt MATH identifier
1019.62047

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 60E05: Distributions: general theory 62E17: Approximations to distributions (nonasymptotic)

Keywords
Complex Wishart moments symmetric group irreducible representations Schur polynomials

Citation

Graczyk, Piotr; Letac, Gérard; Massam, Hélène. The complex Wishart distribution and the symmetric group. Ann. Statist. 31 (2003), no. 1, 287--309. doi:10.1214/aos/1046294466. https://projecteuclid.org/euclid.aos/1046294466


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