Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.

Let $m$ be the maximum of the random variables $x_{i}$ $$ m=\max\{x_i:i=1,2,\ldots,n\} $$

What can one say about $m$? Can we at least compute its mean and variance?

More specifically the problem that I'm interested is the following. Consider a triangular array of random variables where the $n$-th row looks like $$ x_{1}^{(n)},x_{2}^{(n)},\ldots,x_{n}^{(n)} $$ and all the random variables are zero mean and Gaussian. Moreover, $$ \mathbb{Var}(x_{i}^{(n)})=1 \quad \text{for all $1\leq i\leq n$} $$ and $$ \mathbb{Var}(x_{i}^{(n)}x_{j}^{(n)})=\sigma_{ij}(n)\to 0\quad \text{as $n$ increases for $i\neq j$.} $$

Is there anything that can be said about the behavior of $m$ asymptotically?

Thanks!