# Maximum of Correlated Gaussian Random Variables

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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There is no closed form, in general, and the moments can be quite difficult to calculate. Monte Carlo methods are often used in this context. Question: Is there any more structure in the problem you're actually interested in, for example, a certain pattern for the covariance matrix? – cardinal Apr 24 '11 at 15:04
Actually, I'm interested in asymptotics as $n\to\infty$ and in the case where there is correlation but it is very small. In other words, $\sigma_{ij}<<\sigma_{ii}$ for $i\neq j$. – ght Apr 24 '11 at 15:13
Ok. That's a good start. What is your definition of $\sigma_{ij} \ll \sigma_{ii}$ for $i \neq j$? For example, is it something like $\sigma_{ii} = 1$ and $|\sigma_{ij}| < \rho$ for all $i \neq j$? Or, something even stronger? Is there a "base" problem you are trying to generalize from? Maybe that would help pin things down a bit more. – cardinal Apr 24 '11 at 15:24
In my case $\sigma_{ii}=1$ and $\sigma_{ij}=\sigma_{ij}(n)\to 0$ as $n\to\infty$ for $i\neq j$. – ght Apr 24 '11 at 15:27
I know this sounds pedantic, but can you explain the notation further? I think any answer would depend on this information. I can imagine $\sigma_{ij}(n) \to 0$ meaning several things. If you're observing a sequence of random variables, then for fixed $(i,j)$, $\sigma_{ij}$ is constant. If you're observing, say, the rows of a triangular array as your "sequence", then the correlations could be changing even for fixed $(i,j)$. Also, do you know the rate at which the correlations are converging to zero, e.g., $O(n^{-1/2})$? – cardinal Apr 24 '11 at 15:33