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?