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Let $\{X_1,X_2,\ldots,X_n\}$ be jointly Gaussian random variables of zero mean and variance $1$ with covariance matrix $K$. Let $Y=\max\{X_i\,:\,i=1,\ldots,n\}$.

In the case the variables are also independent ($K=I_{n}$) is a known result that $ \mathbb{P}\Big[\lim_{n\to\infty}(2\log(n))^{-1/2}Y=1\Big]=1. $

My question is: Is it true that for a general covariance matrix (with diagonal entries equal to one) then $ \mathbb{P}\Big[\lim_{n\to\infty}(2\log(n))^{-1/2}Y\leq 1\Big]=1? $ I'm only interested in the asymptotic behavior as $n$ increases.

Thanks!

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    @leonbly: fair enough there was a typo. Here is the reference http://www.springerlink.com/content/lt45q14201550468/2012-04-10

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