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Slepian's lemma

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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 ,

the following inequality holds for all real numbers :

or equivalently,

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 , it holds for any real number that

History

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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

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  • 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.