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In [[probability theory]], '''Slepian's lemma''' (1962), named after [[David Slepian]], is a Gaussian comparison inequality. It states that for Gaussian random variables <math>X = (X_1,\dots,X_n)</math> and <math>Y = (Y_1,\dots,Y_n)</math> in <math>\mathbb{R}^n</math> satisfying <math>E[X] = E[Y] = 0</math> and 

:<math>E[X_i^2]=E[Y_i^2], i=1,\dots,n, \ E[X_iX_j] \le E[Y_i Y_j] </math> for <math>i \neq j</math>, 

the following inequality holds for all real numbers <math>u_1,...,u_n</math>:

:<math>P[X_1 \le u_1, \dots, X_n \le u_n] \le P[Y_1 \le u_1, \dots, Y_n \le u_n ] </math>, 

While this intuitive-seeming result is true for Gaussian processes, it is not in general true for other random variables&mdash;not even those with expectation 0.

As a corollary, if <math>(X_t)_{t \ge 0}</math> is a centered [[Stationary process|stationary]] Gaussian process such that <math>E[X_0X_t] \geq 0</math> for all t, it holds for any real number c that

:<math>P\left[\sup_{t \in [0,T+S]} X_t \leq c\right] \ge P\left[\sup_{t \in [0,T]} X_t \leq c\right] P\left[\sup_{t \in [0,S]} X_t \leq c\right], \quad T,S > 0 </math>.

==History==
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 reproven in several different forms.

==References==
* 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.
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[[Category:Lemmas]]