Annals of Probability

Moment inequalities for functions of independent random variables

Stéphane Boucheron, Olivier Bousquet, Gábor Lugosi, and Pascal Massart

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A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003) 1583–1614], and is based on a generalized tensorization inequality due to Latała and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147–168]. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-derive classical inequalities including Rosenthal and Kahane–Khinchine-type inequalities for sums of independent random variables, moment inequalities for suprema of empirical processes and moment inequalities for Rademacher chaos and U-statistics. Some of these corollaries are apparently new. In particular, we generalize Talagrand’s exponential inequality for Rademacher chaos of order 2 to any order. We also discuss applications for other complex functions of independent random variables, such as suprema of Boolean polynomials which include, as special cases, subgraph counting problems in random graphs.

Article information

Ann. Probab., Volume 33, Number 2 (2005), 514-560.

First available in Project Euclid: 3 March 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60C05: Combinatorial probability 28A35: Measures and integrals in product spaces
Secondary: 05C80: Random graphs [See also 60B20]

Moment inequalities concentration inequalities empirical processes random graphs


Boucheron, Stéphane; Bousquet, Olivier; Lugosi, Gábor; Massart, Pascal. Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005), no. 2, 514--560. doi:10.1214/009117904000000856.

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