Skip to main content
SpringerLink
Log in

On the Problem of Reversibility of the Entropy Power Inequality

  • Conference paper
  • First Online:
Limit Theorems in Probability, Statistics and Number Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 42))

Abstract

As was shown recently by the authors, the entropy power inequality can be reversed for independent summands with sufficiently concave densities, when the distributions of the summands are put in a special position. In this note it is proved that reversibility is impossible over the whole class of convex probability distributions. Related phenomena for identically distributed summands are also discussed.

2010 Mathematics Subject Classification. 60F05.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Artstein-Avidan, V. Milman, Y. Ostrover, The M-ellipsoid, symplectic capacities and volume. Comment. Math. Helv. 83(2), 359–369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Bobkov, M. Madiman, Dimensional behaviour of entropy and information. C. R. Acad. Sci. Paris Sér. I Math. 349, 201–204 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Bobkov, M. Madiman, Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262(7), 3309–3339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Bobkov, M. Madiman, The entropy per coordinate of a random vector is highly constrained under convexity conditions. IEEE Trans. Inform. Theor. 57(8), 4940–4954 (2011)

    Article  MathSciNet  Google Scholar 

  5. S.G. Bobkov, Large deviations and isoperimetry over convex probability measures. Electron. J. Probab. 12, 1072–1100 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. S.G. Bobkov, On Milman’s ellipsoids and M-position of convex bodies, in Concentration, Functional Inequalities and Isoperimetry. Contemporary Mathematics, vol. 545 (American Mathematical Society, Providence, 2011), pp. 23–34

    Google Scholar 

  7. S.G. Bobkov, G.P. Chistyakov, F. Götze, Entropic instability of Cramer’s characterization of the normal law, in Selected Works of Willem van Zwet. Sel. Works Probab. Stat. (Springer, New York, 2012), pp. 231–242 (2012)

    Google Scholar 

  8. S.G. Bobkov, G.P. Chistyakov, F. Götze, Stability problems in Cramér-type characterization in case of i.i.d. summands. Teor. Veroyatnost. i Primenen. 57(4), 701–723 (2011)

    Google Scholar 

  9. C. Borell, Complements of Lyapunov’s inequality. Math. Ann. 205, 323–331 (1973)

    MathSciNet  MATH  Google Scholar 

  10. C. Borell, Convex measures on locally convex spaces. Ark. Math. 12, 239–252 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Borell, Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)

    Article  MathSciNet  Google Scholar 

  12. J. Bourgain, B. Klartag, V.D. Milman, Symmetrization and isotropic constants of convex bodies, in Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Mathematics, vol. 1850 (Springer, Berlin, 2004), pp. 101–115

    Google Scholar 

  13. H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Costa, T.M. Cover, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality. IEEE Trans. Inform. Theor. IT-30, 837–839 (1984)

    Article  MathSciNet  Google Scholar 

  15. A. Dembo, T. Cover, J. Thomas, Information-theoretic inequalities. IEEE Trans. Inform. Theor. 37(6), 1501–1518 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. B. Klartag, V.D. Milman, Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Koenig, N. Tomczak-Jaegermann, Geometric inequalities for a class of exponential measures. Proc. Am. Math. Soc. 133(4), 1213–1221 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Madiman, I. Kontoyiannis, The Ruzsa divergence for random elements in locally compact abelian groups. Preprint (2013)

    Google Scholar 

  19. M. Madiman, On the entropy of sums, in Proceedings of the IEEE Information Theory Workshop, Porto, Portugal (IEEE, New York, 2008)

    Google Scholar 

  20. H.P. McKean Jr., Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21, 343–367 (1966)

    Article  MathSciNet  Google Scholar 

  21. V.D. Milman, An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces. C. R. Acad. Sci. Paris Sér. I Math. 302(1), 25–28 (1986)

    MathSciNet  MATH  Google Scholar 

  22. V.D. Milman, Entropy point of view on some geometric inequalities. C. R. Acad. Sci. Paris Sér. I Math. 306(14), 611–615 (1988)

    MathSciNet  MATH  Google Scholar 

  23. V.D. Milman, Isomorphic symmetrizations and geometric inequalities, in Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Mathematics, vol. 1317 (Springer, Berlin, 1988), pp. 107–131

    Google Scholar 

  24. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94 (Cambridge University Press, Cambridge, 1989)

    Google Scholar 

  25. A. Prékopa, Logarithmic concave measures with applications to stochastic programming. Acta Sci. Math. Szeged 32, 301–316 (1971)

    MathSciNet  MATH  Google Scholar 

  26. C.A. Rogers, G.C. Shephard, The difference body of a convex body. Arch. Math. (Basel) 8, 220–233 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  27. C.E. Shannon, A mathematical theory of communication. Bell System Tech. J. 27, 379–423, 623–656 (1948)

    MathSciNet  Google Scholar 

  28. A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Contr. 2, 101–112 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Szarek, D. Voiculescu, Shannon’s entropy power inequality via restricted Minkowski sums, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1745 (Springer, Berlin, 2000), pp. 257–262

    Google Scholar 

Download references

Acknowledgements

Sergey G. Bobkov was supported in part by the NSF grant DMS-1106530. Mokshay M. Madiman was supported in part by the NSF CAREER grant DMS-1056996.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Vincent Hall 228, 206 Church St SE, Minneapolis, MN, 55455, USA

    Sergey G. Bobkov

  2. Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE, 19716, USA

    Mokshay M. Madiman

Authors
  1. Sergey G. Bobkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mokshay M. Madiman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey G. Bobkov .

Editor information

Editors and Affiliations

  1. Mathematics Faculty, Ruhr-University Bochum, Universitätstr. 150, Bochum, 44780, Germany

    Peter Eichelsbacher

  2. Mathematics Faculty, University of Bielefeld, Universitätsstr. 25, Bielefeld, 33615, Germany

    Guido Elsner

  3. Mathematics Faculty, University of Bielefeld, Universitätsstr. 25, Bielefeld, 33615, Germany

    Holger Kösters

  4. Institute for Mathematical Statistics, University of Münster, Einsteinstrasse 62, Münster, 48149, Germany

    Matthias Löwe

  5. Mathematics Institute, University of München, Theresienstr. 39, München, 80333, Germany

    Franz Merkl

  6. Centre for Mathematics, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85747, Germany

    Silke Rolles

Additional information

Dedicated to Friedrich Götze on the occasion of his sixtieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bobkov, S.G., Madiman, M.M. (2013). On the Problem of Reversibility of the Entropy Power Inequality. In: Eichelsbacher, P., Elsner, G., Kösters, H., Löwe, M., Merkl, F., Rolles, S. (eds) Limit Theorems in Probability, Statistics and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36068-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation