Moment inequalities for functions of independent random variables

By Stéphane Boucheron, Olivier Bousquet, Gábor Lugosi, Pascal Massart. 2005

204 citations recorded by the source corpus.

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 Latala 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.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e2e28df95c62a3309560c2f7900aced9a036ffc6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

A note on the Hanson-Wright inequality for random vectors with dependencies (2014, methodology): “A NOTE ON THE HANSON-WRIGHT INEQUALITY 3 [4], whereas the Rademacher case was studied by Talagrand [30] and Bousquet-BoucheronLugosi-Massart [9].”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, methodology): “This follows from simple moment computations or one could use the Efron-Stein inequality, see [17] or equivalently Proposition E.”
A high-dimensional Wilks phenomenon (2011, background): “Taking advantage on moment estimates for general functions of independent random variables derived in (Boucheron et al. 2005b), it is possible to show that the unnormalized empirical excess risk satisfies a Bernstein-like moment inequality where the variance term is of the order of the model dimensi”
On concentration of self-bounding functions (2009, methodology): “For a more concrete class of applications, consider a convex function g defined on a bounded hyperrectangle, say [0, 1 ]n.”
Ranking and empirical minimization of U-statistics (2006, methodology): “Thus, we have In order to bound the moments of the random variable A = E (cid:6) Z (cid:2) (cid:6) , we apply Corol-lary 3 of [9].”
Gaussian approximation of suprema of empirical processes (2012, background): “1), due essentially to the recent work of [3], complemented with a new “local” maximal inequality for the expected supremum of the empirical process that extends the work of [58] (Theorem 5.”
Rejoinder: Local Rademacher complexities and oracle inequalities in risk minimization (2006, methodology): “We use the versions of these inequalities proved by Bousquet [13] and Klein [24] (see [11] for some other relevant inequalities).”
Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings (2015, unknown): “The lemma is essentially due to [2], Theorem 12.”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “…their asymptotic normality under sub-Weibull martingale differences have important implications for concentration results related to functions of independent random variables, which in turn are useful for dependent data (Wu, 2005); see Boucheron et al. (2005) for more applications in this regard.”
Sharp nonasymptotic bounds on the norm of random matrices with independent entries (2014, methodology): “On the other hand, by the Rosenthal-type inequality of [6], Theorem 8, we have ∥∥∥∑ j X2 ij ∥∥∥ α logn ≤ (1 + δ) ∑”
Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels (2014, background): “The following result contains [40, Th. 5] as a special case: Theorem IV.2: Suppose that the assumptions of Theorem IV.1 hold, and that the function is differentiable.”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, methodology): “2 Conditional randomization tests Next we introduce the conditional randomization test (CRT) framework of Candès et al. (2018a). This framework is designed for settings where the conditional distribution of Y given X is known or can be accurately inferred from a large out-of-sample data set.”
A tail inequality for suprema of unbounded empirical processes with applications to Markov chains (2007, methodology): “These moment estimates first appeared in the context of tail inequalities for U -statistics and were later used in statistics, in model selection.”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “tly two of us simpliﬁed the proof of (3) and characterized all measures on Rsatisfying the same property [12] (such a criterion for log-Sobolev already existed, thanks to Bobkov and Go¨tze [15]). See [20] for other developements. Inequalities (3) above are part of a more general family denoted Φ-Sob”