Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On the rate of convergence in the entropic central limit theorem
Download PDF
Download PDF
  • Published: 02 February 2004

On the rate of convergence in the entropic central limit theorem

  • Shiri Artstein1,
  • Keith M. Ball2,
  • Franck Barthe3 &
  • …
  • Assaf Naor4 

Probability Theory and Related Fields volume 129, pages 381–390 (2004)Cite this article

  • 446 Accesses

  • 42 Citations

  • 3 Altmetric

  • Metrics details

Abstract.

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

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Artstein, S., Ball, K., Barthe, F., Naor, A.: Solution of Shannon’s Problem on the Monotonicity of Entropy. Submitted, 2002

  2. Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités XIX, number 1123 in Lect. Notes in Math., Springer, 1985, pp. 179–206

  3. Ball, K., Barthe, F., Naor, A.: Entropy jumps in the presence of a spectral gap. Duke Math. J. 119 (1), 41–63 (2003)

    MATH  Google Scholar 

  4. Barron, A.R.: Entropy and the central limit theorem. Ann. Probab. 14, 336–342 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Barron, A.R., Johnson, O.: Fisher information inequalities and the central limit theorem. Preprint, ArXiv:math.PR/0111020

  6. Blachman, N.M.: The convolution inequality for entropy powers. IEEE Trans. Info. Theory 2, 267–271 (1965)

    Article  Google Scholar 

  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. 314–328

  8. Carlen, E.A., Soffer, A.: Entropy production by block variable summation and central limit theorem. Commun. Math. Phys. 140 (2), 339–371 (1991)

    MATH  Google Scholar 

  9. Csiszar, I.: Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen. Publications of the Mathematical Institute, Hungarian Academy of Sciences, VII, Series A, 137–157 (1962)

  10. Kullback, S.: A lower bound for discrimination information in terms of variation. IEEE Trans. Info. Theory 4, 126–127 (1967)

    Article  Google Scholar 

  11. Linnik, Ju.V.: An information theoretic proof of the central limit theorem with lindeberg conditions. Theory Probab. Appl. 4, 288–299 (1959)

    MATH  Google Scholar 

  12. Pinsker, M.: Information and information stability of random variables and processes. Holden-Day, San Francisco, 1964

  13. Shannon, C.E., Weaver, W.: The mathematical theory of communication. University of Illinois Press, Urbana, IL, 1949

  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. Reidel

  15. Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Info. Control 2, 101–112 (1959)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Shiri Artstein

  2. Department of Mathematics, University College London, Gower Street, London, WC1 6BT, United Kingdom

    Keith M. Ball

  3. Institut de Mathematiques, Laboratoire de Statistiques et Probabilites-CNRS UMR, C5583, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex, France

    Franck Barthe

  4. Theory Group, Microsoft Research, One Microsoft Way, Redmond WA Redmond WA Group, 98052-6399, USA

    Assaf Naor

Authors
  1. Shiri Artstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keith M. Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Franck Barthe
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Assaf Naor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiri Artstein.

Additional information

Mathematics Subjects Classification (2000):94A17; 60F05

Supported in part by the EU Grant HPMT-CT-2000-00037, The Minkowski center for Geometry and the Israel Science Foundation.

Supported in part by NSF Grant DMS-9796221.

Supported in part by EPSRC Grant GR/R37210.

Supported in part by the BSF, Clore Foundation and EU Grant HPMT-CT-2000-00037.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Artstein, S., Ball, K., Barthe, F. et al. On the rate of convergence in the entropic central limit theorem. Probab. Theory Relat. Fields 129, 381–390 (2004). https://doi.org/10.1007/s00440-003-0329-4

Download citation

  • Received: 27 September 2003

  • Revised: 15 November 2003

  • Published: 02 February 2004

  • Issue Date: July 2004

  • DOI: https://doi.org/10.1007/s00440-003-0329-4

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Entropy
  • Linear Combination
  • Limit Theorem
  • Central Limit
  • Central Limit Theorem
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature