Maximum of Gaussian Random Variables - MathOverflow

Maximum of Gaussian Random Variables

2011-04-30T01:49:34Z

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 by Igor Rivin for Maximum of Gaussian Random Variables

2011-04-30T03:57:49Z

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 by Or Zuk for Maximum of Gaussian Random Variables

2011-04-30T15:53:01Z

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 by Karadi for Maximum of Gaussian Random Variables

2011-07-07T11:50:32Z

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