Suppose $Z_1$ and $Z_2$ are bivariate normal, marginally standardized, and has correlation $\rho$. Fix $z_0>0$.
I am interested in finding an estimate of the rate of convergence the probability $P(Z_1<z_0, Z_2>z_0)$ to $0$ as $\rho\uparrow 1$. Is such type of result known?