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Proof of a conjecture of M.L. Eaton on the characteristic function of the Wishart distribution. (English) Zbl 0728.62053
Summary: Let m (\(\geq 2)\) be a positive integer; \(I_ m\) be the \(m\times m\) identity matrix; and \(\Sigma\) and A be symmetric \(m\times m\) matrices, where \(\Sigma\) is positive definite.
By proving that the function \(\phi_{\alpha}(A)=| I_ m-2iA\Sigma |^{-\alpha}\) is a characteristic function only if \[ \alpha \in \{0,1/2,1,3/2,...,(m-2)/2\}\cup [(m-1)/2,\infty), \] we establish a conjecture of M. L. Eaton [Multivariate statistics. A vector space approach. (1983; Zbl 0587.62097)]. A similar result is established for the rank 1 noncentral Wishart distribution and is conjectured to also be valid for any greater rank.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E10 Characteristic functions; other transforms
62H10 Multivariate distribution of statistics
60E07 Infinitely divisible distributions; stable distributions
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