Prove partial uncorrelation implies independence, and vice versa in the special case of Gaussian random variables

Question

Show that if $X, Y $and $Z$ are (univariate) Gaussian random variables then $\rho_{X,Y |Z} = 0$ if and only if $X\perp Y | Z$ (X is independent of Y given Z).

Answer 1

I believe you also require the further assumption that $X,Y,Z$ are jointly Gaussian for this to be true - certainly, you will need to assume this if you want to apply the fact that you quoted. Then you will form the vector $(X,Y,Z)$ and apply the fact you quoted with $p=3$ and $A=\{1\}, B=\{2\}, C=\{3\}$. It is very explicit if you write out what the condition $\rho_{X,Y\mid Z}=0$ means in terms of the covariance matrix, but I will not do this for you since I am under the impression you are asking a homework question.