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On a Nonsymmetric Version of the Khinchine-Kahane Inequality

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Stochastic Inequalities and Applications

Part of the book series: Progress in Probability ((PRPR,volume 56))

Abstract

We prove a new version of the Khinchine—Kahane inequality in which Bernoulli random variables no longer need to be symmetric. The constant in the inequality is optimal up to some universal factor. The proof uses hypercontractive methods and the optimal hypercontractivity constant for a mean-zero Bernoulli random variable is found. A simple observation generalizing Pisier’s Rademacher projection norm estimate is added.

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References

  1. W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102 (1975), 159–182.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bonami, Construction d’opérateurs de convolution sur le groupe D ∞, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A655—A657

    Google Scholar 

  3. A. Bonami, Étude des coefficients de Fourier des fonctions de L P (G) Ann. Inst. Fourier (Grenoble), 20 1970 fasc. 2 (1971), 335–402

    Google Scholar 

  4. C. Boren, On the integrability of Banach space valued polynomials, in Séminaire de Probabilités,XIII (Univ. Strasbourg,Strasbourg, 1977/78), 1–3, Lecture Notes in Math. 721, Springer, Berlin, 1979

    Google Scholar 

  5. P. Diaconis, L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Prob. 6 (1996), 695–750

    Article  MathSciNet  MATH  Google Scholar 

  6. U. Haagerup, The best constants in the Khinchine inequality, Studia Math. 70 (1981), 231–283

    MathSciNet  Google Scholar 

  7. Y. Higuchi, N. Yoshida, Analytic conditions and phase transition for Ising models, Lecture Notes in Japanese (1995)

    Google Scholar 

  8. J.-P. Kahane, Sur les sommes vectorielles Σ ± u n ,C. R. Acad. Sci. Paris 259 (1964), 2577–2580

    MathSciNet  MATH  Google Scholar 

  9. A. Khintchine, Über dyadische Brüche, Math. Z. 18 (1923), 109–116

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Kindler and S. Safra, Noise-Resistant Boolean-Functions are Juntas (personal communication)

    Google Scholar 

  11. W. Krakowiak and J. Szulga, Hypercontraction principle and random multilinear forms, Probab. Theory Related Fields 77 (1988), 325–342

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Kwapień, R. Latala and K. Oleszkiewicz, Comparison of moments of sums of independent random variables and differential inequalities, J. Funct. Anal. 136 (1996), 258–268

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Kwapień and J. Szulga, Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, Ann. Probab. 19 (1991), 369–379

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Kwapień and W. A. Woyczynski, Random series and stochastic integrals: single and multiple, Probability and its Applications, Birkh00E4user, 1992

    Book  MATH  Google Scholar 

  15. R. Latala, Estimation of moments of sums of independent real random variables, Ann. Probab. 25 (1997), 1502–1513

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Latala and K. Oleszkiewicz, On the best constant in the Khinchine-Kahane inequality Studia Math. 109 (1994), 101–104

    MathSciNet  MATH  Google Scholar 

  17. R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, 147–168, Lecture Notes in Math. 1745, Springer, 2000

    Google Scholar 

  18. K. Oleszkiewicz, Comparison of moments via Poincaré-type inequality, in Advances in stochastic inequalities (Atlanta, GA, 1997), 135–148, Contemp. Math. 234, Amer. Math. Soc., 1999

    Google Scholar 

  19. G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Math. 94, Cambridge University Press, 1989

    Book  MATH  Google Scholar 

  20. S. B. Steckin, On the best lacunary systems of functions, (in Russian) Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 357–366

    MathSciNet  Google Scholar 

  21. S. J. Szarek, On the best constants in the Khinchine inequality, Studia Math. 58 (1976), 197–208

    MathSciNet  MATH  Google Scholar 

  22. M. Talagrand On Russo’s approximate zero-one law, Ann. Prob. 22 (1994), 1576–587

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Whittle, Bounds for the moments of linear and quadratic forms in independent variables, Teor. Verojatnost. i Primenen. 5 (1960), 331–335

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

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

    Krzysztof Oleszkiewicz

Authors
  1. Krzysztof Oleszkiewicz
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Editor information

Editors and Affiliations

  1. Department of Mathematics, U-3009, University of Connecticut, Storrs, CT, 06268, USA

    Evariste Giné

  2. Laboratoire d’Analyse et de Mathématiques Appliquées CNRS UMR 8050, Université de Paris XII, 94010, Créteil Cedex, France

    Christian Houdré

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, 30332, USA

    Christian Houdré

  4. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007, Barcelona, Spain

    David Nualart

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Oleszkiewicz, K. (2003). On a Nonsymmetric Version of the Khinchine-Kahane Inequality. In: Giné, E., Houdré, C., Nualart, D. (eds) Stochastic Inequalities and Applications. Progress in Probability, vol 56. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8069-5_11

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  • DOI: https://doi.org/10.1007/978-3-0348-8069-5_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9428-9

  • Online ISBN: 978-3-0348-8069-5

  • eBook Packages: Springer Book Archive

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