[Paper Review] Stein's method and stochastic analysis of Rademacher functionals

Ivan Nourdin, Giovanni Peccati+1 more|Oct 16, 2008

Random Matrices and Applications4 citations

TL;DR

This paper develops a novel application of Stein's method combined with discrete Malliavin calculus to derive explicit bounds for the normal approximation of functionals of infinite Rademacher sequences. By constructing explicit exchangeable pairs via chaos expansions and using contraction norms of kernels, it establishes sharp CLT bounds for multilinear forms, weighted infinite 2-runs, and multiple integrals over sparse sets—providing an alternate, refined proof of a result by Blei and Janson on sparse multiple integrals with explicit error rates.

ABSTRACT

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.

Motivation & Objective

• To develop a framework for normal approximation of functionals of infinite Rademacher sequences using Stein's method and discrete Malliavin calculus.
• To derive explicit bounds in the central limit theorem (CLT) for functionals not necessarily of partial sum form.
• To construct explicit exchangeable pairs for finite Rademacher functionals via chaotic decomposition, enabling application of existing Stein method results.
• To provide a new, refined proof of a combinatorial CLT for multiple integrals over sparse sets, originally established by Blei and Janson.
• To extend the applicability of Stein’s method to functionals of i.i.d. Rademacher sequences indexed by arbitrary discrete sets, independent of ordering.

Proposed method

• Utilizes a discrete version of Malliavin calculus, including discrete divergence and gradient operators, to analyze functionals of Rademacher sequences.
• Employs the chaotic decomposition of square-integrable functionals into orthogonal multiple integrals to express the functional in terms of kernels.
• Constructs an explicit exchangeable pair for any functional depending on finitely many Rademacher variables using the chaos expansion, satisfying a linearity condition in conditional expectation.
• Defines contraction norms of kernels on discrete sets using counting measures, which serve as key components in bounding the normal approximation error.
• Applies the results of Reinert and Röllin [34] on exchangeable pairs to derive explicit bounds in the Wasserstein distance.
• Combines the framework with Mossel et al. [21] to extend results to general i.i.d. sequences with finite third moments, enabling comparison between Rademacher and Gaussian functionals.

Experimental results

Research questions

• RQ1 Can explicit bounds in the normal approximation of functionals of infinite Rademacher sequences be derived without relying on classical exchangeable pairs?
• RQ2 How can discrete Malliavin calculus be used to construct exchangeable pairs for finite Rademacher functionals?
• RQ3 What role do contraction norms of kernels play in bounding the convergence rate in CLTs for functionals in a fixed chaos?
• RQ4 Can the sparseness condition in Blei and Janson’s CLT for multiple integrals be naturally derived from contraction norms rather than moment computations?
• RQ5 To what extent can the framework be generalized to i.i.d. sequences indexed by arbitrary discrete sets, independent of ordering?

Key findings

• Explicit bounds in the Wasserstein distance are derived for the normal approximation of functionals of infinite Rademacher sequences, expressed in terms of contraction norms of the kernels in the chaotic decomposition.
• For functionals in a fixed chaos, the bounds are given by the operator norms of the contraction kernels, which generalize results from Gaussian and Poisson chaos settings.
• The paper provides a new, refined proof of a CLT for multiple integrals over sparse sets, with explicit error bounds that match or improve upon previous results.
• The sparseness condition required for convergence in Blei and Janson’s theorem emerges naturally from the contraction norm structure, rather than from moment-based martingale arguments.
• An explicit upper bound is derived for the distance between the distribution of a Rademacher functional and a standard Gaussian, expressed as $ B_1 rac{|F_N^ atural|^{1/2}}{|F_N|} + B_2 ig[ ext{max}_j rac{|F_{Nj}^*|}{|F_N|}ig]^{1/4} $, where $ F_N^ atural $ and $ F_{Nj}^* $ are combinatorial quantities related to the functional's structure.
• The framework is extended to general i.i.d. sequences with finite third moments, showing that the Rademacher CLT results can be transferred to the Gaussian case via a comparison result from Mossel et al. [21].