Maximum of Gaussian Random Variables

Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.

Let $m$ be the maximum of the random variables $x_{i}$ $$m=\max\{x_i:i=1,2,\ldots,n\}$$

What can one say about $m$? Can we at least compute its mean and variance?

More specifically the problem that I'm interested is the following. Consider a triangular array of random variables where the $n$-th row looks like $$x_{1}^{(n)},x_{2}^{(n)},\ldots,x_{n}^{(n)}$$ and all the random variables are zero mean and Gaussian. Moreover, $$\mathbb{Var}(x_{i}^{(n)})=1 \quad \text{for all 1\leq i\leq n}$$ and $$\mathbb{Var}(x_{i}^{(n)}x_{j}^{(n)})=\sigma_{ij}(n)\to 0\quad \text{as n increases for i\neq j.}$$

Is there anything that can be said about the behavior of $m$ asymptotically?

Thanks!

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If the correlations decay fast enough $\sigma_{ij}(n) = o(1/\log n)$, then the asymptotic distribution of the maximum is the same as if the variables were independent (i.e. the standard Gumbel distribution) - see:

Limit Theorems for the Maximum Term in Stationary Sequences, S.M. Berman (Ann. Math. Statist. 1964) http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177703551

and also: On the asymptotic joint distribution of the sum and maximum of stationary normal random variables H.C. Ho and T. Hsing (Journal of applied probability, 1996). http://www.jstor.org/pss/3215271

For the general case (correlations decay slower or not at all) I don't know of exact results for the limit, but there is a work showing how to compute bounds on the expectation for finite $n$:

Useful Bounds on the Expected Maximum of Correlated Normal Variables, A.M. Ross (2003) http://people.emich.edu/aross15/q/papers/bounds_Emax.pdf

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Thanks for the references Or! I will take a careful look at these papers. –  ght May 1 '11 at 12:33

C.E.Clark's paper on Maximum of a finite set of random variables provides a reasonable closed form approximation. You can always write max(x1,x2,x3) as max(x1,max(x2,x3)). Clark's paper basically uses this fact and tries to create a chain for finite number of variables

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See: On the distribution of the maximum of random variables, by J. Galambos (Annals of Math. Stat, 1972). For your convenience, the pdf is here.

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@Igor: Thanks for the paper! However, I'm not so sure that this helps with the triangular array problem though. What do you think? It might be possible that I'm missing something. –  ght Apr 30 '11 at 13:48
I did not think about the triangular array thing, but would think that it could be gotten out the reference with some work. Maybe @Or Zuk'z Ross reference does it? I did not check yet... –  Igor Rivin Apr 30 '11 at 15:55