Indefinite covariance matrix, multivariate normal distribution, how to compute the variance of $X+Y+Z$

Question

If $X$, $Y$, and $Z$ each follows the standard normal distribution, and each pair of them follows a bivariate normal distribution with a correlation coefficient of $-\frac{2}{3}$. Now we have $$ \operatorname*{var}\left( \begin{bmatrix} X\\ Y\\ Z \end{bmatrix} \right) =% \begin{bmatrix} 1 & -\frac{2}{3} & -\frac{2}{3}\\ -\frac{2}{3} & 1 & -\frac{2}{3}\\ -\frac{2}{3} & -\frac{2}{3} & 1 \end{bmatrix} =\mathbf{V}% $$ However, $\mathbf{V}$ is an indefinite matrix, not a positive semidefinite matrix. Thus, $\mathbf{V}$ can not be a covariance matrix. If $\mathbf{V}$ is the covariance matrix $$ \operatorname*{var}(X+Y+Z)=% \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}% \begin{bmatrix} 1 & -\frac{2}{3} & -\frac{2}{3}\\ -\frac{2}{3} & 1 & -\frac{2}{3}\\ -\frac{2}{3} & -\frac{2}{3} & 1 \end{bmatrix}% \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} =-1<0 $$ Is it possible to compute the variance of $X+Y+Z$? My guess is that they do not have a joint multivariate normal distribution.

Answer 1

Any covariance matrix $\mathbf{V}$ is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). The correlation matrix is $$ \mathbf{R}=\mathbf{D}^{-\frac{1}{2}}\mathbf{VD}^{-\frac{1}{2}}% $$ where $\mathbf{D}=\mathrm{diag}(\mathbf{V})$ is the matrix of the diagonal elements of $\mathbf{V}$. Thus, the correlation matrix $\mathbf{R}$ must be positive semi-definite. If $\rho_{XY}=\rho_{XZ}=-\frac{2}{3}$, there must be $-\frac{1}{9}\leq\rho_{YZ}\leq1$, not $-1\leq\rho_{YZ}\leq1$, we can never take $\rho_{YZ}=-\frac{2}{3}$.