Some Distribution Problems in the Multivariate Complex Gaussian Case
Some Distribution Problems in the Multivariate Complex Gaussian Case
Let $\mathbf{X}_1: p \times n$ and $\mathbf{X}_2: p \times n$ be real random variables having the joint density function \begin{equation*}\tag{1.1} (2\pi)^{-pn}|\mathbf{\Sigma}_0|^{-\frac{1}{2}n} \exp \{-\frac{1}{2} \operatorname{tr}\mathbf{\Sigma}_0^{-1}(\mathbf{X} - \mathbf{\nu})(\mathbf{X} - \mathbf{\nu})'\},\quad - \infty \leqq \mathbf{X} \leqq \infty\end{equation*} where \begin{equation*}\mathbf{X} = \binom{\mathbf{X}_1}{\mathbf{X}_2},\quad \mathbf{\Sigma}_o = \begin{pmatrix}\mathbf{\Sigma}_1 & -\mathbf{\Sigma}_2 \\ \mathbf{\Sigma}_2 & \mathbf{\Sigma}_1\end{pmatrix},\quad \nu = \begin{pmatrix}\mathbf{\mu}_1 …