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The complex Wishart distribution and the symmetric group. (English) Zbl 1019.62047

Summary: 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 \(\mathbb{E}(\text{exp trace}(\theta S))=(\det(I_r -\sigma \theta))^{-p}\), where \(\sigma\) and \(\sigma^{-1} -\theta\) are in \(\Omega\), and where \(p=1,2,\dots, r-1\) or \(p>r-1\). We compute all moments of \(S\) and \(S^{-1}\). The techniques involve in particular the use of the irreducible characters of the symmetric group.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
20B30 Symmetric groups
20C30 Representations of finite symmetric groups
60E05 Probability distributions: general theory

Software:

GAP
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References:

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