What are the conditions for the covariances of a multivariate gaussian? For a 3 dimensional multivariate gaussian I have 3 + 3 + 3 free variables, in the mean, the standard deviation and the covariances. For simplicity we assume a standardized multivariate gaussian (mean = 0, standard deviation = 1). If I know two of the three covariances, $\rho_{12}$ and $\rho_{13}$, what can I say about $\rho_{23}$?
I presume that there would be more or less simple inequalities about maximum and minimum values for $\rho_{23}$. However, the only thing I can come up with is solving for positive definitenes, which is kind of complicated.
Additionally, how would this inequality change if I change the dimension of the gaussian?
 A: You need the covariance matrix to be positive semidefinite.
As StubbornAtom says, if the covariance matrix is $\Sigma=\begin{pmatrix}1&\rho_{12}&\rho_{13}\\\rho_{12}&1&\rho_{23}\\\rho_{13}&\rho_{23}&1\end{pmatrix}$, i.e. also the correlation matrix, then this amounts to requiring $\det \Sigma \ge 0$ since all the principal minors are $1$ or $1-\rho^2_{ij} \ge 0$.
If you know $\rho_{12}$ and $\rho_{13}$ then this corresponds to an interval for $\rho_{23}$:
$$\rho_{12}\rho_{13} - \sqrt{(1-\rho_{12}^2)(1-\rho_{13}^2)} \le  \rho_{23} \le \rho_{12}\rho_{13} + \sqrt{(1-\rho_{12}^2)(1-\rho_{13}^2)}$$
This could be generalised to remove the marginal standard normal assumption,  by substituting for the correlations by using covariances and variances. Starting with $\Sigma=\begin{pmatrix}\sigma^2_1&\sigma_{12}&\sigma_{13}\\\sigma_{12}&\sigma^2_2&\sigma_{23}\\\sigma_{13}&\sigma_{23}&\sigma^2_3\end{pmatrix}$
and knowing $\sigma^2_1,\sigma^2_{2},\sigma^2_3,\sigma_{12},\sigma_{13}$, you would get
$$\frac{\sigma_{12}\sigma_{13} - \sqrt{(\sigma^2_{1}\sigma^2_2-\sigma_{12}^2)(\sigma^2_{1}\sigma^2_3-\sigma_{13}^2)}}{\sigma^2_1} \le  \sigma_{23} \le \frac{\sigma_{12}\sigma_{13} + \sqrt{(\sigma^2_{1}\sigma^2_2-\sigma_{12}^2)(\sigma^2_{1}\sigma^2_3-\sigma_{13}^2)}}{\sigma^2_1}$$
