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Let $Y,X$ be jointly normally distributed and assume that they are highly correlated. I'm interested in knowing what happens to the survival function as the variables become perfectly correlated. Specifically, I'm interested in this instance:

$$ \lim_{\rho_{YX} \rightarrow1} \Pr[Y>y,X>\mu_X] $$

I tried getting around the integrals but did not get anywhere. I plotted numerically different examples and found that it does seem to converge.

Below is the plot of $\Pr[Y>y,X>\mu_X]$ (blue) and $\Pr[Y>y]$ (dashed red) for a bivariate normal distribution with $\mu_Y=\mu_X=0$, $\sigma_Y=\sigma_X=1$ and $\rho_{YX}=0.999$. As it can be seen, the former converges to the latter for $y>0$. How can I prove this and get the general result?

Example of survival function at limit

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up vote 2 down vote accepted

When $\mu_X=\mu_Y=0$ and $\sigma_X=\sigma_Y=1$, the random vector $(X,Y)$ can be realized as $(X,Y)=(\varrho Y+\sqrt{1-\varrho^2}Z,Y)$ where $Z$ is standard normal and independent of $Y$. Then, $A_\varrho=[Y\gt y,X\gt0]$ is $A_\varrho=[Y\gt y,Z\gt-a(\varrho)Y]$ with $a(\varrho)=\varrho/\sqrt{1-\varrho^2}$. Since $a(\varrho)\to+\infty$ when $\varrho\to+1$, $[Z\gt-a(\varrho)y]\to\Omega$ if $y\gt0$ and $[Z\gt-a(\varrho)y]\to\varnothing$ if $y\lt0$. Thus, $P[A_\varrho]\to P[B]$ where $B=[Y\gt y,Y\gt0]=[Y\gt\max(y,0)]$. This explains the blue curve.

In the general case, $(X,Y)=(\mu_X+\sigma_X\varrho T+\sigma_X\sqrt{1-\varrho^2}Z,\mu_Y+\sigma_YT)$ where $Z$ and $T$ are standard normal and independent. Then, $A_\varrho=[Y\gt y,X\gt\mu_X]$ is $$ A_\varrho=[T\gt(y-\mu_Y)/\sigma_Y,Z\gt-a(\varrho)T]. $$ Since $a(\varrho)\to+\infty$ when $\varrho\to+1$, $[Z\gt-a(\varrho)t]\to\Omega$ if $t\gt0$ and $[Z\gt-a(\varrho)t]\to\varnothing$ if $t\lt0$. Thus, $P[A_\varrho]\to P[B]$ where $B=[T\gt(y-\mu_Y)/\sigma_Y,T\gt0]$, that is, $B=[T\gt\max((y-\mu_Y)/\sigma_Y,0)]$. This yields a translation by $\mu_Y$ of a dilation by $\sigma_Y$ of the blue curve.

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