Suppose $y\sim N(0,\Sigma)$ is an $n-$dimensional vector. I'm interested in an upper bound for $\Pr(\max\_{1\leq i\leq n} y\_i > k)$ for $k$ large. I know a little about $\Sigma$: $\sigma\_{ii}=\sigma\_{jj}$ for any $i, j$ and $\sigma\_{ij}\geq 0$. I can also bound the pairwise correlations from above, but it's rather uninformative (something like $(m-1)/m$ where $m$ is large in my application). But that's about it. 


There are all kinds of results about similar probabilities for smooth Gaussian processes, but I can't find much about regular old random vectors. Any pointers?



Bounding exceedance probabilities for correlated normal variables
Slepian's inequality allows you to dominate the probability for the case you ask about by the same with $\Sigma=\sigma\_{11} I$.


