The Annals of Probability

Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution

Shyamal Das Peddada and Donald St. P. Richards

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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.

Article information

Source
Ann. Probab. Volume 19, Number 2 (1991), 868-874.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176990455

Digital Object Identifier
doi:10.1214/aop/1176990455

Mathematical Reviews number (MathSciNet)
MR1106290

Zentralblatt MATH identifier
0728.62053

JSTOR
links.jstor.org

Subjects
Primary: 62E15: Exact distribution theory
Secondary: 62H10: Distribution of statistics 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

Citation

Peddada, Shyamal Das; Richards, Donald St. P. Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution. Ann. Probab. 19 (1991), no. 2, 868--874. doi:10.1214/aop/1176990455. http://projecteuclid.org/euclid.aop/1176990455.


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See also

  • Acknowledgment of Prior Result: Shyamal D. Peddada, Donald St. P. Richards. Acknowledgment of Priority: Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution. Ann. Probab., Volume 20, Number 2 (1992), 1107--1107.