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 and in satisfying and

for ,

the following inequality holds for all real numbers :

,

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 is a centered stationary Gaussian process such that for all t, it holds for any real number c that

.