Borell–TIS inequality

In mathematics and probability, the Borell–TIS inequality is a result bounding the probability of a deviation of the uniform norm of a centred Gaussian stochastic process above its expected value. The result is named for Christer Borell and its independent discoverers Boris Tsirelson, Ildar Ibragimov, and Vladimir Sudakov. The inequality has been described as “the single most important tool in the study of Gaussian processes.”[1]

Statement

Let ${\displaystyle T}$ be a topological space, and let ${\displaystyle \{f_{t}\}_{t\in T}}$ be a centred (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 ${\displaystyle u>0}$,

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

See also

• Gaussian isoperimetric inequality

References

1. "Gaussian Inequalities". Random Fields and Geometry. New York, NY: Springer New York. 2007. pp. 49–64. doi:10.1007/978-0-387-48116-6_2. ISBN 978-0-387-48116-6.