Probability Theory and Related Fields

, Volume 129, Issue 3, pp 381–390 | Cite as

On the rate of convergence in the entropic central limit theorem

  • Shiri Artstein
  • Keith M. Ball
  • Franck Barthe
  • Assaf Naor


We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.


Entropy Linear Combination Limit Theorem Central Limit Central Limit Theorem 
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  1. 1.
    Artstein, S., Ball, K., Barthe, F., Naor, A.: Solution of Shannon’s Problem on the Monotonicity of Entropy. Submitted, 2002Google Scholar
  2. 2.
    Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités XIX, number 1123 in Lect. Notes in Math., Springer, 1985, pp. 179–206Google Scholar
  3. 3.
    Ball, K., Barthe, F., Naor, A.: Entropy jumps in the presence of a spectral gap. Duke Math. J. 119 (1), 41–63 (2003)zbMATHGoogle Scholar
  4. 4.
    Barron, A.R.: Entropy and the central limit theorem. Ann. Probab. 14, 336–342 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barron, A.R., Johnson, O.: Fisher information inequalities and the central limit theorem. Preprint, ArXiv:math.PR/0111020Google Scholar
  6. 6.
    Blachman, N.M.: The convolution inequality for entropy powers. IEEE Trans. Info. Theory 2, 267–271 (1965)CrossRefGoogle Scholar
  7. 7.
    Brown, L.D.: A proof of the central limit theorem motivated by the Cramer-Rao inequality. In: Kalliampur et al., (eds.), Statistics and Probability: Essays in Honor of C. R. Rao, Amsterdam, North-Holland, 1982, pp. 314–328Google Scholar
  8. 8.
    Carlen, E.A., Soffer, A.: Entropy production by block variable summation and central limit theorem. Commun. Math. Phys. 140 (2), 339–371 (1991)zbMATHGoogle Scholar
  9. 9.
    Csiszar, I.: Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen. Publications of the Mathematical Institute, Hungarian Academy of Sciences, VII, Series A, 137–157 (1962)Google Scholar
  10. 10.
    Kullback, S.: A lower bound for discrimination information in terms of variation. IEEE Trans. Info. Theory 4, 126–127 (1967)CrossRefGoogle Scholar
  11. 11.
    Linnik, Ju.V.: An information theoretic proof of the central limit theorem with lindeberg conditions. Theory Probab. Appl. 4, 288–299 (1959)zbMATHGoogle Scholar
  12. 12.
    Pinsker, M.Open image in new window: Information and information stability of random variables and processes. Holden-Day, San Francisco, 1964Google Scholar
  13. 13.
    Shannon, C.E., Weaver, W.: The mathematical theory of communication. University of Illinois Press, Urbana, IL, 1949Google Scholar
  14. 14.
    Shimizu, R.: On Fisher’s amount of information for location family. In: Patil et al., (eds.), A modern course on statistical distributions in scientific work, Boston, MA, 1974. D. ReidelGoogle Scholar
  15. 15.
    Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Info. Control 2, 101–112 (1959)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Shiri Artstein
    • 1
  • Keith M. Ball
    • 2
  • Franck Barthe
    • 3
  • Assaf Naor
    • 4
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsUniversity College LondonLondonUnited Kingdom
  3. 3.Institut de MathematiquesLaboratoire de Statistiques et Probabilites-CNRS UMRToulouse cedexFrance
  4. 4.Theory GroupMicrosoft ResearchUSA

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