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

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.