Jump to content

Borell–TIS inequality

From Wikipedia, the free encyclopedia

In mathematics and probability, the Borell–TIS inequality is a result bounding the probability of a deviation of the uniform norm of a centered 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

[edit]

Let be a topological space, and let be a centered (i.e. mean zero) Gaussian process on , with

almost surely finite, and let

Then[1] and are both finite, and, for each ,

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

,

and so by symmetry

.

See also

[edit]

References

[edit]
  1. ^ a b c "Gaussian Inequalities". Random Fields and Geometry. Springer Monographs in Mathematics. 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.