Suppose you have $X,Y\sim\mathcal{N}(0,1)$ jointly Gaussian distributed with correlation coefficient $\rho$. I am looking for a convenient formula of the following expression: $$\mathbb{P}(\{|X|>1\}\cap\{|Y|>1\}).$$ In particular, I am interested in the limit $\rho\rightarrow0$. If $\rho=0$, we know that $$\mathbb{P}(\{|X|>1\}\cap\{|Y|>1\})=\mathbb{P}(\{|X|>1\})\cdot\mathbb{P}(\{|Y|>1\})=\mathbb{P}(\{|X|>1\})^2$$ which can be expressed using the error function. But as of right now, I am not even able to show this convergence, because the integrals involved in calculating the left hand side for $\rho>0$ are too unwieldy.
It would be great if I could find a rate of convergence of $$\mathbb{P}(\{|X|>1\}\cap\{|Y|>1\})-\mathbb{P}(\{|X|>1\})^2\xrightarrow{\rho\rightarrow 0} 0$$
Thanks!
I have found a rate of convergence that is sufficient to me. It is the case that $$F(\rho):=\mathbb{P}(\{|X|>1\}\cap\{|Y|>1\})-\mathbb{P}(\{|X|>1\})^2\leq L|\rho|$$ for some constant $L>0$. My proof of this is quite lengthy but it comes down to writing the function as \begin{equation*} F(\rho)=\int_{M}f_{0,\Sigma_\rho}(x_1,x_2)-f_{0,\Sigma_0}(x_1,x_2)d(x_1,x_2) \end{equation*}
on $I:=[-\rho_0,\rho_0]$, $\rho_0<1$. Here, $f_{0,\Sigma_\rho}$ is the two dimensional Gaussian pdf with mean (0,0) and covariance matrix \begin{equation*} \Sigma_\rho=\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} \end{equation*} If we can show that $F$ is continuously differentiable with bounded derivative on $I$, we are done and $L$ is just that bound. The lengthy part is showing that for all $\rho\in I$, the functions $\partial_\rho f_{0,\Sigma_\rho}$ are uniformly bounded by an integrable function $g:\mathbb R^2\rightarrow\mathbb R_+$ so that we may exchange differentiation and integral in $\partial_\rho F(\rho)$.
AM, Berlin