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 $$\operatorname E[X] = \operatorname E[Y] = 0$$,


 * $$\operatorname E[X_i^2]= \operatorname E[Y_i^2], \quad i=1,\dots,n, \text{ and } \operatorname E[X_iX_j] \le \operatorname E[Y_i Y_j] \text{ for } i \neq j.$$

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


 * $$\Pr\left[\bigcap_{i=1}^n \{X_i \le u_i\}\right] \le \Pr\left[\bigcap_{i=1}^n \{Y_i \le u_i\}\right], $$

or equivalently,


 * $$\Pr\left[\bigcup_{i=1}^n \{X_i > u_i\}\right] \ge \Pr\left[\bigcup_{i=1}^n \{Y_i > u_i\}\right]. $$

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

As a corollary, if $$(X_t)_{t \ge 0}$$ 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


 * $$\Pr\left[\sup_{t \in [0,T+S]} X_t \leq c\right] \ge \Pr\left[\sup_{t \in [0,T]} X_t \leq c\right] \Pr \left[\sup_{t \in [0,S]} X_t \leq c\right], \quad T,S > 0. $$

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.