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!
Best Answer
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