Correlation of IID normal RV's

Question

If we know that $corr(X,Y)=\rho$ and $Y,Z$ are independent and also $X,Y,Z\sim N(0,1)$ then can we deduce anything about $corr(X,Z)$?

Answer 1

Clearly $\text{corr}(Y,Z)=0$. If $\text{corr}(X,Y)=\rho^{\,}_{XY}$ and $\text{corr}(X,Y)=\rho^{\,}_{XY}$ then the covaraince matrix is $$\begin{bmatrix} 1 & \rho^{\,}_{XY} & \rho^{\,}_{XZ}\\ \rho^{\,}_{XY} & 1 & 0 \\ \rho^{\,}_{XZ} & 0 & 1\end{bmatrix}$$

A necessary condition for this to be positive semidefinite is that its determinant is non-negative, i.e. $$1 - \rho_{XY}^2 - \rho_{XZ}^2 \ge 0$$ which implies $$-\sqrt{1-\rho_{XY}^2} \le \rho_{XZ}^{\,} \le \sqrt{1-\rho_{XY}^2}$$

In general, that is not sufficient, but it is here since the other principal minors have determinants of $1$ or $1-\rho_{XY}^2$ or $1-\rho_{XZ}^2$, which are all non-negative.

As examples of all such values in this intervals, suppose we have another standard normal random variable $W$ independent of $Y$ and $Z$ so we can construct $X= \rho^{\,}_{XY}Y +\rho^{\,}_{XZ}Z +\sqrt{1 - \rho_{XY}^2 - \rho_{XZ}^2}W$. This will give the desired correlation matrix for $X,Y,Z$, at least so long as $\sqrt{1 - \rho_{XY}^2 - \rho_{XZ}^2}$ is real.