Suppose two random variables $X$ and $Y$, and their covariance could be $E(XY)$ if we simply assume their expectations are zero.
Now, we take the conditional expectation of both: $\xi = E(X\mid Z)$ and $\eta = E(Y\mid Z)$, and their covariance is $E(\xi\eta)$
I am interested in the quantity $(E(XY))^2 - (E(\xi\eta))^2$
This guy is non-negative. My question is that, is there an upper bound? How about the case when $X=Y$? What conditions of $EX^2$ and $EY^2$ should be to get an upper bound? Thanks.
If $Z$ is constant, or more generally if $X$ and $Z$ are independent and $Y$ and $Z$ are independent, you'd have $\xi = E(X|Z) = E(X) = 0$ and $\eta = E(Y|Z) = E(Y) = 0$, so $E(\xi \eta) = 0$. Then the question reduces to getting an upper bound on $E(XY)^2$. For that, you could use Cauchy-Schwarz.