Maximum of Gaussian Random Variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:17:31Z http://mathoverflow.net/feeds/question/63490 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63490/maximum-of-gaussian-random-variables Maximum of Gaussian Random Variables ght 2011-04-30T01:49:34Z 2011-07-07T11:50:32Z <p>Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.</p> <p>Let $m$ be the maximum of the random variables $x_{i}$ $$<br> m=\max\{x_i:i=1,2,\ldots,n\}$$</p> <p>What can one say about $m$? Can we at least compute its mean and variance?</p> <p>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 $$\mathbb{Var}(x_{i}^{(n)}x_{j}^{(n)})=\sigma_{ij}(n)\to 0\quad \text{as n increases for i\neq j.}$$ </p> <p>Is there anything that can be said about the behavior of $m$ asymptotically? </p> <p>Thanks!</p> http://mathoverflow.net/questions/63490/maximum-of-gaussian-random-variables/63495#63495 Answer by Igor Rivin for Maximum of Gaussian Random Variables Igor Rivin 2011-04-30T03:57:49Z 2011-04-30T03:57:49Z <p>See: On the distribution of the maximum of random variables, by J. Galambos (Annals of Math. Stat, 1972). For your convenience, the pdf is <a href="http://dl.dropbox.com/u/5188175/galambos.pdf" rel="nofollow">here.</a></p> http://mathoverflow.net/questions/63490/maximum-of-gaussian-random-variables/63537#63537 Answer by Or Zuk for Maximum of Gaussian Random Variables Or Zuk 2011-04-30T15:53:01Z 2011-04-30T15:53:01Z <p>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: </p> <p>Limit Theorems for the Maximum Term in Stationary Sequences, S.M. Berman (Ann. Math. Statist. 1964) <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aoms/1177703551" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aoms/1177703551</a></p> <p>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). <a href="http://www.jstor.org/pss/3215271" rel="nofollow">http://www.jstor.org/pss/3215271</a></p> <p>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 $n$: </p> <p>Useful Bounds on the Expected Maximum of Correlated Normal Variables, A.M. Ross (2003) <a href="http://people.emich.edu/aross15/q/papers/bounds_Emax.pdf" rel="nofollow">http://people.emich.edu/aross15/q/papers/bounds_Emax.pdf</a></p> http://mathoverflow.net/questions/63490/maximum-of-gaussian-random-variables/69705#69705 Answer by Karadi for Maximum of Gaussian Random Variables Karadi 2011-07-07T11:50:32Z 2011-07-07T11:50:32Z <p>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</p>