# Inequality of covariances between a bivariate normal vector and its indicator functions

Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that \begin{equation} |\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|? \end{equation} With $1\{Z_1\leq u\}$ we denote the indicator function of the set $\{\omega\in\Omega|Z_1(\omega)\leq u\}$, $u\in\mathbb{R}$ fixed.
Thanks!

• I think you can show that the correlation of any 2 functions F(Z 1 ),H(Z 2 ) is less than ρ= correlation os Z 1 ,Z 2 . I have this result mentally filed under 'goblin's inequality', which google is not helpful with. A short proof can be got by representing F(Z 1 ),H(Z 2 ) as stochastic integrals of correlated weiner processes and using the Ito isometry , and can be found in Revuz and Yor. – – mike Jun 11 '13 at 15:51
• Thanks! I found it:) It is called "Gebelein Inequality". Do you know if there exists a comparable but more general result if $Z=(Z_1,Z_2)$ isn't normal? – stroem Jun 12 '13 at 10:07