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!