Let $X_t$, $t \geq 0$, be a centered continuous Gaussian process. Is there a general, useful, bound on the variance of the supremum $\sup_{0\leq t \leq T} X_t$?
1 Answer
By the Borell--Tsirelson--Ibragimov--Sudakov inequality on concentration of the maximum of a centered Gaussian process, for $M:=\sup_{0\le t\le T} X_t$ and all real $x>0$ we have $$P(|M-EM|>x)\le2 e^{-x^2/(2\sigma^2)},$$ where $\sigma^2:=\sup_{0\le t\le T}EX_t^2$. So, the variance of $M$ is $$E(M-EM)^2=\int_0^\infty 2x\,dx\,P(|M-EM|>x) \le\int_0^\infty 2x\,dx\,2 e^{-x^2/(2\sigma^2)}=4\sigma^2.$$
This bound on the variance of $M$ is obviously optimal up to a universal constant factor, because the variance of $M$ equals $\sigma^2$ if $X_t=X_0$ for all $t$.
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$\begingroup$ Can one expect the optimal bound $\sigma^2$ (without the constant 4)? $\endgroup$ Jul 14, 2021 at 8:54
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$\begingroup$ @DmitriRozev : Good question. I do not know an answer to it, though. $\endgroup$ Jul 14, 2021 at 12:16
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$\begingroup$ @DimitriRosev : If your problem admits a bit more structure, you might want to have a look at the book 'Random fields and geometry' by Adler and Taylor (Springer). $\endgroup$ Jul 16, 2021 at 14:47