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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.

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  • $\begingroup$ You should find very good bounds if you look for "chaining argument" and "Fernique-Talagrand". $\endgroup$
    – Martin Hairer
    Commented Feb 28, 2014 at 2:59
  • $\begingroup$ @MartinHairer Thanks Martin, but as far as I can tell the majorizing measures theory talks only about the mean; I'm more interested in the variance. Perhaps I haven't looked enough. $\endgroup$
    – arjun
    Commented Feb 28, 2014 at 12:50

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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).

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  • $\begingroup$ Thanks Ofer, since this is an isoperimetric estimate, do you also know if Talagrand's hypercontractivity based inequality can be used to produce a general improvement? $\endgroup$
    – arjun
    Commented Feb 28, 2014 at 16:46
  • $\begingroup$ +1. But the link is broke. $\endgroup$
    – Hans
    Commented Oct 2, 2018 at 6:56
  • $\begingroup$ en.wikipedia.org/wiki/Borell-TIS_inequality $\endgroup$
    – ofer zeitouni
    Commented Oct 3, 2018 at 2:28

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