Slepian's lemma

In probability theory, Slepian's lemma (1962), named after David Slepian, is a Gaussian comparison inequality. It states that for Gaussian random variables ${\displaystyle X=(X_{1},\dots ,X_{n})}$ and ${\displaystyle Y=(Y_{1},\dots ,Y_{n})}$ in ${\displaystyle \mathbb {R} ^{n}}$ satisfying ${\displaystyle \operatorname {E} [X]=\operatorname {E} [Y]=0}$,

${\displaystyle \operatorname {E} [X_{i}^{2}]=\operatorname {E} [Y_{i}^{2}],\quad i=1,\dots ,n,{\text{ and }}\operatorname {E} [X_{i}X_{j}]\leq \operatorname {E} [Y_{i}Y_{j}]{\text{ for }}i\neq j.}$

the following inequality holds for all real numbers ${\displaystyle u_{1},\ldots ,u_{n}}$:

${\displaystyle \Pr \left[\bigcap _{i=1}^{n}\{X_{i}\leq u_{i}\}\right]\leq \Pr \left[\bigcap _{i=1}^{n}\{Y_{i}\leq u_{i}\}\right],}$

or equivalently,

${\displaystyle \Pr \left[\bigcup _{i=1}^{n}\{X_{i}>u_{i}\}\right]\geq \Pr \left[\bigcup _{i=1}^{n}\{Y_{i}>u_{i}\}\right].}$

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 ${\displaystyle (X_{t})_{t\geq 0}}$ is a centered stationary Gaussian process such that ${\displaystyle \operatorname {E} [X_{0}X_{t}]\geq 0}$ for all ${\displaystyle t}$, it holds for any real number ${\displaystyle c}$ that

${\displaystyle \Pr \left[\sup _{t\in [0,T+S]}X_{t}\leq c\right]\geq \Pr \left[\sup _{t\in [0,T]}X_{t}\leq c\right]\Pr \left[\sup _{t\in [0,S]}X_{t}\leq c\right],\quad T,S>0.}$