Slepian's lemma

In probability theory, Slepian's lemma (1962), named after David Slepian, states that for any two finite-state Gaussian stochastic processes

:X, Y

with non-singular covariance matrices

:$A,\; B$

satisfying

:$a\_\{ii\}=b\_\{ii\},,a\_\{ij\}>b\_\{ij\}$,

the expectation

:E ["f"(X)]

will be greater than

:E ["f"(Y)]

for all bounded, continuous L-superadditive functions "f".

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.

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 reproven 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.