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{1176990455,
author = {Peddada, Shyamal Das and Richards, Donald St. P.},
title = {Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution},
journal = {Ann. Probab.},
volume = {19},
number = {4},
year = {1991},
pages = { 868-874},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990455}
}
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., Tome 19 (1991) no. 4, pp. 868-874. http://gdmltest.u-ga.fr/item/1176990455/