Maximizing $cov(X,Y)$ for given $F_X, F_Y$

Question

Given $F_X$ and $F_Y$ of zero mean and unit variance, what is the maximum value for $cov(X,Y)$? Is it simply $$\rho_{max}=E(F_X^{-1}(U)\cdot F_Y^{-1}(U))$$ for a standard uniform $U$? Is there an alternative representation?

Answer 1

If $X$ is given and $Y$ is allowed to vary over all variables of zero mean and unit covariance, then the maximum covariance is reached for $X=Y,$ i.e., covariance 1.

The covariance of two zero-mean variables is really the inner product of two vectors in a Hilbert space, and your question is about the maximum of the inner product of two unit vectors.