In probability theory, Slepian's lemma (1962), named after David Slepian, is a Gaussian comparison inequality. It states that for Gaussian random variables
and
in
satisfying
,
![{\displaystyle \operatorname {E} [X_{i}^{2}]=\operatorname {E} [Y_{i}^{2}],\quad i=1,\dots ,n,{\text{ and }}\operatorname {E} [X_{i}X_{j}]\leq \operatorname {E} [Y_{i}Y_{j}]{\text{ for }}i\neq j.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72cb860237916943d580ab7719bba4d74fc651b)
the following inequality holds for all real numbers
:
![{\displaystyle \Pr \left[\bigcap _{i=1}^{n}\{X_{i}\leq u_{i}\}\right]\leq \Pr \left[\bigcap _{i=1}^{n}\{Y_{i}\leq u_{i}\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80a0df091217bd945cb711093965fa84f225c9f7)
or equivalently,
![{\displaystyle \Pr \left[\bigcup _{i=1}^{n}\{X_{i}>u_{i}\}\right]\geq \Pr \left[\bigcup _{i=1}^{n}\{Y_{i}>u_{i}\}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57f4d43735e868932db946a32eb58d611d3715b9)
While this intuitive-seeming result is true for Gaussian processes, it is not in general true for other random variables—not even those with expectation 0.
As a corollary, if
is a centered stationary Gaussian process such that
for all
, it holds for any real number
that
![{\displaystyle \Pr \left[\sup _{t\in [0,T+S]}X_{t}\leq c\right]\geq \Pr \left[\sup _{t\in [0,T]}X_{t}\leq c\right]\Pr \left[\sup _{t\in [0,S]}X_{t}\leq c\right],\quad T,S>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e996118382633e82d564d91717bf4d0b1d7c4f)
Slepian's lemma was first proven by Slepian in 1962, and has since been used in reliability theory, extreme value theory and areas of pure probability. It has also been re-proven in several different forms.
- Slepian, D. "The One-Sided Barrier Problem for Gaussian Noise", Bell System Technical Journal (1962), pp 463–501.
- Huffer, F. "Slepian's inequality via the central limit theorem", Canadian Journal of Statistics (1986), pp 367–370.
- Ledoux, M., Talagrand, M. "Probability in Banach Spaces", Springer Verlag, Berlin 1991, pp 75.