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!



Maximum of Gaussian Random Variables
If the correlations decay fast enough $\sigma\_{ij}(n) = o(1/\log n)$, then the asymptotic distribution of the maximum is the same as if the variables were independent (i.e.
the standard Gumbel distribution) - see:


Limit Theorems for the Maximum Term in Stationary Sequences, S.M. Berman (Ann. Math. Statist. 1964)
[Link](https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-35/issue-2/Limit-Theorems-for-the-Maximum-Term-in-Stationary-Sequences/10.1214/aoms/1177703551.full)


and also:
On the asymptotic joint distribution of the sum and maximum of stationary normal random variables H.C. Ho and T. Hsing (Journal of applied probability, 1996).
<https://www.jstor.org/stable/3215271>


For the general case (correlations decay slower or not at all) I don't know of exact results for the limit, but there is a work showing how to compute bounds on the expectation for finite $n$:


Useful Bounds on the Expected Maximum of Correlated Normal Variables, A.M. Ross (2003)
[https://emunix.emich.edu/~aross15/q/papers/bounds\_Emax.pdf](https://emunix.emich.edu/%7Eaross15/q/papers/bounds_Emax.pdf)


