If you have that X and Y are independent random variables and their covariance is equal to zero, then is the $Cov(X^2, Y^2)$ zero as well? I feel like this is too good to be true, but I cannot come up with a conclusion.
[Math] Covariance of X^2 Y^2 when Cov(X,Y) = 0
probabilitystatistics
Best Answer
Yes that is true.
If $X$ and $Y$ are independent it implies their correlation, and therefore covariance are zero (but the converse is not necessarily true).
Assuming this knowledge, then by squaring the RVs separately we are not inducing any new correlational information, and thus they still remain uncorrelated and independent.
In this way we conclude $Cov(X,Y) = Cov(X^2,Y^2) = 0$.