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 $\mathbb^n$ satisfying \operatorname E = \operatorname E = 0,

:\operatorname E _i^2 \operatorname E _i^2 \quad i=1,\dots,n, \text \operatorname E _iX_j\le \operatorname E_i Y_j\text i \neq j.

the following inequality holds for all real numbers $u_1,\ldots,u_n$:

:\Pr\left bigcap_^n \\right\le \Pr\left bigcap_^n \\right

or equivalently,

:\Pr\left bigcup_^n \\right\ge \Pr\left bigcup_^n \\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 $(X_t)_$ is a centered stationary Gaussian process such that \operatorname E _0 X_t\geq 0 for all $t$, it holds for any real number $c$ that

:\Pr\left sup_ X_t \leq c\right\ge \Pr\left sup_ X_t \leq c\right\Pr \left sup_ 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.

Lemmas

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