# 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!

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

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

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.

• @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... Apr 30 '11 at 15:55

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

I can provide you concentration inequality around the mean, in particular the decay is exponential et reflects the correct size of the variance. It is important to notice that Classical concentration inequality for Lipschitz functions can not be applied.

As a reference see :

http://perso.math.univ-toulouse.fr/ktanguy/files/2012/04/Article-3-brouillon.pdf

Kevin Tanguy