Let X, Y, and Z are three random variables. Intuitively, I think that it is

impossibleto 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 equalityimpossible.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 ρ>0 we may have Cor(X,Y)=ρ,Cor(Y,Z)=ρ and Cor(X,Z)=0(1).

My questions are:

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

**Answer**

The correlation matrix needs to be positive semi-definite with non-negative eigenvalues. The eigenvalues of the correlation matrix are the solutions of

|1−λρρρ1−λ0ρ01−λ|=(1−λ)((1−λ)2−2ρ2))=0

so the eigenvalues are 1 and 1±√2ρ. These are all non-negative for

−1√2≤ρ≤1√2.

**Attribution***Source : Link , Question Author : TrungDung , Answer Author : Jarle Tufto*