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April, 1991 Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution
Shyamal Das Peddada, Donald St. P. Richards
Ann. Probab. 19(2): 868-874 (April, 1991). DOI: 10.1214/aop/1176990455

Abstract

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 $\alpha \in \{0, \frac{1}{2}, 1,\frac{3}{2},\ldots,(m - 2)/2\} \cup \lbrack(m - 1)/2, \infty)$, we establish a conjecture of Eaton. A similar result is established for the rank 1 noncentral Wishart distribution and is conjecture to also be valid for any greater rank.

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Shyamal Das Peddada. Donald St. P. Richards. "Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution." Ann. Probab. 19 (2) 868 - 874, April, 1991. https://doi.org/10.1214/aop/1176990455

Information

Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0728.62053
MathSciNet: MR1106290
Digital Object Identifier: 10.1214/aop/1176990455

Subjects:
Primary: 62E15
Secondary: 60D05 , 62H10

Keywords: Characteristic function , decomposability , Delphic semigroup , Infinite divisibility , Laguerre polynomial of matrix argument , orthogonal group , Schur function , Wishart distribution , zonal polynomial

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.19 • No. 2 • April, 1991
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