Skip to main content
SpringerLink
Log in

Gaussian concentration for a class of spherically invariant measures

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Concentration and logarithmic Sobolev inequalities are derived for a class of multidimensional probability distributions, including spherically invariant log-concave measures. Bibliography: 17 titles.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. G. Bobkov, “Spectral gap and concentration for some spherically symmetric probability measures,” Lect. Notes Math. 1807 (2003), 37–43.

    MathSciNet  Google Scholar 

  2. M. Gromov, V. D. Milman, “A topological application of the isoperimetric inequality,” Am. J. Math. 105, 843–854 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. A. Borovkov, S. A. Utev, “On an inequality and related characterization of the normal distribution,” Probab. Theory Appl. 28, 209–218 (1983).

    MATH  MathSciNet  Google Scholar 

  4. S. G. Bobkov, “Remarks on Gromov–Milman’s inequality” [in Russian], Vestn. Syktyvkar Univ. Ser. 1 3, 15–22 (1999).

    MATH  MathSciNet  Google Scholar 

  5. A. Prékopa, “Logarithmic concave measures with application to stochastic programming.” Acta Sci. Math. (Szeged) 32, 301–316 (1971).

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. V. D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lect. Notes Math. 1200, Springer, Berlin (1986).

  8. M. Ledoux, The Concentration of Measure Phenomenon, Am. Math. Soc., Providence, RI (2001).

    MATH  Google Scholar 

  9. R. E. Barlow, A. W. Marshall, F. Proshan, “Properties of probability distributions with monotone hazard rate,” Ann. Math. Stat. 34, 375–389 (1963).

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Klartag, “A central limit theorem for convex sets,” Invent. Math. 168, No. 1, 91–131 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  12. L. Lovász, M. Simonovits, “Random walks in a convex body and an improved volume algorithm,” Random Struct. Algorithms 4, No. 3, 359–412 (1993).

    Article  MATH  Google Scholar 

  13. O. Guédon, “Kahane–Khinchine type inequalities for negative exponent,” Mathematika, 46, No. 1, 165–173 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. G. Bobkov, “Isoperimetric and analytic inequalities for log-concave probability distributions,” Ann. Probab. 27, No. 4, 1903–1921 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  15. C. E. Mueller, F. B. Weissler, “Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere,” J. Funct. Anal. 48, 252–283 (1992).

    Article  MathSciNet  Google Scholar 

  16. S. Karlin, F. Proshan, R. E. Barlow, “Moment inequalities of Polya frequency functions,” Pacific J. Math. 11, 1023–1033 (1961).

    MATH  MathSciNet  Google Scholar 

  17. S. G. Bobkov, I. Gentil, M. Ledoux, “Hypercontractivity of Hamilton-Jacobi equations,” J. Math. Pures Appl. 80, No. 7, 669–696 (2001).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, USA

    S. G. Bobkov

Authors
  1. S. G. Bobkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. G. Bobkov.

Additional information

Translated from Problems in Mathematical Analysis 46, April 2010, pp. 45–56.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bobkov, S.G. Gaussian concentration for a class of spherically invariant measures. J Math Sci 167, 326–339 (2010). https://doi.org/10.1007/s10958-010-9922-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-010-9922-0

Keywords

Search

Navigation