An extremal property of the normal distribution, with a discrete analog

Erwan Hillion*, Oliver Johnson, Adrien Saumard

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
105 Downloads (Pure)


We give a new proof, using only the Brascamp–Lieb inequality, of the fact that the Gaussian measure is the only strong log-concave measure having a strong log-concavity parameter equal to its covariance matrix. We unify the continuous and discrete settings by also giving a similar characterization of the Poisson measure in the discrete case, using “Chebyshev’s other inequality”. We briefly discuss how these results relate to Stein and Stein–Chen methods for Gaussian and Poisson approximation, and to the Bakry–Émery calculus.
Original languageEnglish
Pages (from-to)181-186
Number of pages6
JournalStatistics and Probability Letters
Early online date25 Sept 2018
Publication statusPublished - Feb 2019


  • Brascamp–Lieb inequality
  • Characterization of laws
  • Chebychev's other inequality
  • Log-concavity
  • Normal distribution
  • Poisson distribution


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