Let $x_{1},x_{2},\cdots,x_{n}$ and $y_{1},y_{2},\cdots,y_{n}$ be correlated normal random variables the covariance between two arbitrary random variables is known. In other words, let $X=\left[x_{1},\cdots,x_{n}\right]^{T}\in\mathbb{R}^{n}$ and $Y=\left[y_{1},\cdots,y_{n}\right]^{T}\in\mathbb{R}^{n}$, we have \begin{equation} \left[\begin{array}{c} X\\ Y \end{array}\right]\sim\mathcal{N}\left(\left[\begin{array}{c} \mu_{X}\\ \mu_{Y} \end{array}\right],\left[\begin{array}{cc} \Sigma_{X} & \Sigma_{XY}\\ \Sigma_{YX} & \Sigma X \end{array}\right]\right) \end{equation}
where the mean vectors $\mu_{X}$ and $\mu_{Y}$ and the covariance matrices $\Sigma_{X}$, $\Sigma_{Y}$, $\Sigma_{XY}=\Sigma_{YX}$ are known. The problem is to find the distribution of a random variable $z$ defined by \begin{equation} z=\max_{1\leq k\leq n}x_{k}+\max_{1\leq k\leq n}y_{k} \end{equation} Could you please give me any suggestion?