I am trying to understand for bivariate normal variables, when their correlation increases, can we show in certian sense the distribution is concentrated on the 45 degree line.
For example, let $F(X,Y,\rho)$ be the CDF of a bivariate normal distribution with mean $0$ and covariance matrix $\begin{pmatrix} 1, \rho\\\rho,1\end{pmatrix}$. For $\rho\geq 0$ and $x\leq y$, can we show $$\frac{\partial^2 \log F(x,y,\rho)}{\partial x \, \partial \rho}\geq 0 \quad ?$$