Skip to main content
SpringerLink
Log in

Moments of Unconditional Logarithmically Concave Vectors

  • Chapter
  • First Online:
Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2050))

Abstract

We derive two-sided bounds for moments of linear combinations of coordinates of unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.

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 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight 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. F. Barthe, O. Guédon, S. Mendelson, A. Naor, A probabilistic approach to the geometry of the l p n-ball. Ann. Probab. 33, 480–513 (2005)

    Google Scholar 

  2. S.G. Bobkov, F.L. Nazarov, in On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1807 (Springer, Berlin, 2003), pp. 53–69

    Google Scholar 

  3. E.D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114, 303–309 (1995)

    MathSciNet  MATH  Google Scholar 

  4. U. Haagerup, The best constants in the Khintchine inequality. Studia Math. 70, 231–283 (1982)

    MathSciNet  MATH  Google Scholar 

  5. P. Hitczenko, Domination inequality for martingale transforms of a Rademacher sequence. Israel J. Math. 84, 161–178 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Huet, Spectral gap for some invariant log-concave probability measures. Mathematika 57, 51–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Kannan, L. Lovász, M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13, 541–559 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theor. Relat. Fields 145, 1–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Latała, Gaussian approximation of moments of sums of independent symmetric random variables with logarithmically concave tails. IMS Collect. 5, 37–42 (2009)

    Google Scholar 

  11. R. Latała, Weak and strong moments of random vectors. Banach Center Publ. 95, 115–121 (2011)

    Article  Google Scholar 

  12. R. Latała, J.O. Wojtaszczyk, On the infimum convolution inequality. Studia Math. 189, 147–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Lis, Gaussian approximation of moments of sums of independent random variables. Bull. Polish Acad. Sci. Math. 60, 77–90 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. S.J. Montgomery-Smith, The distribution of Rademacher sums. Proc. Am. Math. Soc. 109, 517–522 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Nazarov, M. Sodin, A. Volberg, The geometric KLS lemma, dimension-free estimates for the distribution of values of polynomials, and distribution of zeroes of random analytic functions (Russian). Algebra i Analiz 14, 214–234 (2002); Translation in St. Petersburg Math. J. 14, 351–366 (2003)

    Google Scholar 

  17. G. Paouris, in Ψ 2 -Estimates for Linear Functionals on Zonoids. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1807 (Springer, Berlin, 2003), pp. 211–222

    Google Scholar 

  18. M. Pilipczuk, J.O. Wojtaszczyk, Negative association property for absolute values of random variable equidistributed on generalized Orlicz balls. Positivity 12, 421–474 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Q.-M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Probab. 13, 343–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. J.O. Wojtaszczyk, A simpler proof of the negative association property for absolute values of measures tied to generalized Orlicz balls. Bull. Pol. Acad. Sci. Math. 57, 41–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Part of this work was done at the Newton institute for Mathematical Sciences in Cambridge (UK) during the program Discrete Analysis. Research partially supported by MNiSW Grant no. N N201 397437 and the Foundation for Polish Science.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    Rafał Latała

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warszawa, Poland

    Rafał Latała

Authors
  1. Rafał Latała
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafał Latała .

Editor information

Editors and Affiliations

  1. Tel-Aviv, 69978, Israel

    Bo'az Klartag

  2. Technion Machine Learning Center, Technion, Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Shahar Mendelson

  3. Fac. Exact Sciences, School of Mathematical Sciences, University of Tel Aviv, Tel Aviv, 69978, Israel

    Vitali D. Milman

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Latała, R. (2012). Moments of Unconditional Logarithmically Concave Vectors. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_17

Download citation

Publish with us

Policies and ethics

Search

Navigation