Borell–TIS inequality

Let ${\displaystyle T}$ be a topological space, and let ${\displaystyle \{f_{t}\}_{t\in T}}$ be a centered (i.e. mean zero) Gaussian process on ${\displaystyle T}$, with

${\displaystyle \|f\|_{T}:=\sup _{t\in T}|f_{t}|}$

almost surely finite, and let

${\displaystyle \sigma _{T}^{2}:=\sup _{t\in T}\operatorname {E} |f_{t}|^{2}.}$

Then[1] ${\displaystyle \operatorname {E} (\|f\|_{T})}$ and ${\displaystyle \sigma _{T}}$ are both finite, and, for each ${\displaystyle u>0}$,

${\displaystyle \operatorname {P} {\big (}\|f\|_{T}>\operatorname {E} (\|f\|_{T})+u{\big )}\leq \exp \left({\frac {-u^{2}}{2\sigma _{T}^{2}}}\right).}$

Another related statement which is also known as the Borell-TIS inequality[1] is that, under the same conditions as above,

${\displaystyle \operatorname {P} {\big (}\sup _{t\in T}f_{t}>\operatorname {E} (\sup _{t\in T}f_{t})+u{\big )}\leq \exp {\bigg (}{\frac {-u^{2}}{2\sigma _{T}^{2}}}{\bigg )}}$,

and so by symmetry

${\displaystyle \operatorname {P} {\big (}|\sup _{t\in T}f_{t}-\operatorname {E} (\sup _{t\in T}f_{t})|>u{\big )}\leq 2\exp {\bigg (}{\frac {-u^{2}}{2\sigma _{T}^{2}}}{\bigg )}}$.

• Gaussian isoperimetric inequality