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Let $x_{1},x_{2},\ldots, x_{n}$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij}^{2})_{1\leq i,j\leq n}$. In other words, $ \mathrm{cov}(x_i,x_j)=\sigma_{ij}^{2} $ for all $i,j\in \{1,2,\ldots,n\}$.

Let $ m=\max \{x_{i}:i=1,\ldots,n\} $ be the maximum of the $x_{i}$. Is it known what is the distribution of $m$? Can we at least compute its mean and variance?

Are there asymptotic results as $n\to\infty$, at least for the weak correlation case ($\sigma_{ij}\ll \sigma_{ii}$ for $i\neq j$)? For instance, assume that $\sigma_{ii}=1$ and $\sigma_{ij}=\sigma_{ij}(n)\to 0$ as $n\to\infty$ for $i\neq j$.

Thanks!

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    Triangular array. The rates are complicated.2011-04-24

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