## 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.

### MSC:

 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

Zbl 0587.62097
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