Is this possible that Cor(X,Y)=0.99Cor(X, Y)=0.99, Cor(Y,Z)=0.99Cor(Y, Z)=0.99 but Cor(X,Z)=0Cor(X, Z)=0?

Question 1

Let $X$ , $Y$ , and $Z$ are three random variables. Intuitively, I think that it is impossible to have $Cor(X, Y)=0.99$ , $Cor(Y, Z)=0.99$ but $Cor(X, Z)=0$ . My intuitive thought is that $X$ and $Z$ are nearly linearly correlated to $Y$ . Hence, they are more or less linearly correlated, which makes the last equality impossible.

I pose this question because of the question and the comments (include my comments) here.

In general, as some others point out, I agree that it is possible that for some $\rho>0$ we may have $Cor(X, Y)=\rho, Cor(Y, Z)=\rho \mbox{ and } Cor(X, Z)=0 \qquad (1).$

My questions are:

Do you think that (1) is wrong when $\rho$ is close to 1, e.g., 0.99?
If (1) is wrong when $\rho$ is close to 1, what is the maximum value of $\rho$ so that (1) can be correct?

Question 2

The correlation matrix needs to be positive semi-definite with non-negative eigenvalues. The eigenvalues of the correlation matrix are the solutions of
$\left| \begin{matrix} 1-\lambda & \rho & \rho \\ \rho & 1-\lambda & 0 \\ \rho & 0 & 1-\lambda \end{matrix} \right| =(1-\lambda)\big((1-\lambda)^2-2\rho^2)\big) =0$
so the eigenvalues are $1$ and $1\pm\sqrt{2}\rho$ . These are all non-negative for
$-\frac1{\sqrt{2}} \le \rho \le \frac1{\sqrt{2}}.$

Is this possible that Cor(X,Y)=0.99Cor(X, Y)=0.99, Cor(Y,Z)=0.99Cor(Y, Z)=0.99 but Cor(X,Z)=0Cor(X, Z)=0?

Answer

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