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 $X=(X_{1},\dots ,X_{n})$ and $Y=(Y_{1},\dots ,Y_{n})$ in $\mathbb {R} ^{n}$ satisfying $\operatorname {E} [X]=\operatorname {E} [Y]=0$,

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

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

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

or equivalently,

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

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

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

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 re-proven 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.