Let $\{X\_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X\_1,\ldots,X\_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some symmetric, positive-definite matrix. 


I'm trying to estimate the variance of $f(X)$, and was wondering if someone could give me references for this. This should fall under standard extreme value theory, but things are quite correlated here. 



Variance of maximum of mixture of gaussians
A general bound on the variance is given by the Borell (Tsirelson-Ibragimov-Sudakov) inequality, see 
 <http://webee.technion.ac.il/people/adler/borell.pdf>


Without more structure on A I don't think it can be improved (think of the case where A has rank 1 to convince yourself of that).


