On a Nonsymmetric Version of the Khinchine-Kahane Inequality

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.

[1]  A. Khintchine.  Über dyadische Brüche , 1923 .

[2]  P. Whittle,et al.  Bounds for the Moments of Linear and Quadratic Forms in Independent Variables , 1960 .

[3]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .

[4]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[5]  S. Szarek.  On the best constants in the Khinchin inequality , 1976 .

[6]  C. Borell.  On the integrability of Banach space valued Walsh polynomials , 1979 .

[7]  U. Haagerup.  The best constants in the Khintchine inequality , 1981 .

[8]  J. Szulga,et al.  Hypercontraction principle and random multilinear forms , 1988 .

[9]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[10]  Stanisław Kwapień,et al.  Hypercontraction Methods in Moment Inequalities for Series of Independent Random Variables in Normed Spaces , 1991 .

[11]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

[12]  M. Talagrand.  On Russo's Approximate Zero-One Law , 1994 .

[13]  R. Latala,et al.  On the best constant in the Khinchin-Kahane inequality , 1994 .

[14]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[15]  Krzysztof Oleszkiewicz,et al.  Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities , 1996 .

[16]  R. Latala.  Estimation of moments of sums of independent real random variables , 1997 .

[17]  Krzysztof Oleszkiewicz,et al.  Comparison of moments via Poincare-type inequality , 1999 .

[18]  R. Latala,et al.  Between Sobolev and Poincaré , 2000, math/0003043.

[19]  S. Safra,et al.  Noise-Resistant Boolean-Functions are Juntas , 2003 .