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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?

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  • $\begingroup$ Please google via: Hartigan : Bounding the maximum of dependent random variables Project Euclid › euclid.ejs $\endgroup$
    – Chee
    Commented Dec 21, 2015 at 5:04

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Slepian's inequality allows you to dominate the probability for the case you ask about by the same with $\Sigma=\sigma_{11} I$.

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  • $\begingroup$ Useful, thanks. I'll leave it open for a bit to see if perhaps I can get anything tighter. Seems unlikely I guess; I know there are some nonzero off diagonal elements in $\Sigma$, but I don't know where or how large they are. $\endgroup$
    – JMS
    Commented Dec 21, 2015 at 23:25

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