Question

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!

Answer 1

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

Answer 2

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.

Answer 3

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