Electronic Journal of Probability

Stein's Method and Stochastic Analysis of Rademacher Functionals

Gesine Reinert, Ivan Nourdin, and Giovanni Peccati

Full-text: Open access


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. As the bounds are given in terms of Malliavin operators, no coupling construction is required. When the functional depends only on the first d coordinates of the Rademacher sequence, a simple sufficient condition for convergence to a normal distribution is derived. For finite quadratic forms, we obtain necessary and sufficient conditions. Although our approach does not require the classical use of exchangeable pairs, when the functional depends only on the first d coordinates of the Rademacher sequence we employ chaos expansion in order to construct an explicit exchangeable pair vector; the elements of the vector relate to the summands in the chaos decomposition and satisfy a linearity condition for the conditional expectation. Among several examples, such as 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.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 55, 1703-1742.

Accepted: 3 November 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F99: None of the above, but in this section 60G50: Sums of independent random variables; random walks 60H07: Stochastic calculus of variations and the Malliavin calculus

Central Limit Theorems Discrete Malliavin operators Normal approximation Rademacher sequences Sparse sets Stein's method Walsh chaos

This work is licensed under aCreative Commons Attribution 3.0 License.


Reinert, Gesine; Nourdin, Ivan; Peccati, Giovanni. Stein's Method and Stochastic Analysis of Rademacher Functionals. Electron. J. Probab. 15 (2010), paper no. 55, 1703--1742. doi:10.1214/EJP.v15-823. https://projecteuclid.org/euclid.ejp/1464819840

Export citation


  • Balakrishnan, N.; Koutras, Markos V. Runs and scans with applications. Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons], New York, 2002. xxii+452 pp. ISBN: 0-471-24892-4
  • Barbour, A. D. Stein's method for diffusion approximations. Probab. Theory Related Fields 84 (1990), no. 3, 297–322.
  • Bentkus, Vidmantas; Jing, Bing-Yi; Zhou, Wang. On normal approximations to $U$-statistics. Ann. Probab. 37 (2009), no. 6, 2174–2199.
  • Blei, Ron. Analysis in integer and fractional dimensions. Cambridge Studies in Advanced Mathematics, 71. Cambridge University Press, Cambridge, 2001. xx+556 pp. ISBN: 0-521-65084-4
  • Blei, Ron; Janson, Svante. Rademacher chaos: tail estimates versus limit theorems. Ark. Mat. 42 (2004), no. 1, 13–29.
  • Chatterjee, Sourav. A new method of normal approximation. Ann. Probab. 36 (2008), no. 4, 1584–1610.
  • Chatterjee, Sourav. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (2009), no. 1-2, 1–40.
  • Chen, Louis H. Y.; Shao, Qi-Man. Stein's method for normal approximation. An introduction to Stein's method, 1–59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
  • Daly, Fraser. Upper bounds for Stein-type operators. Electron. J. Probab. 13 (2008), no. 20, 566–587.
  • de Jong, Peter. A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 (1987), no. 2, 261–277.
  • de Jong, Peter. A central limit theorem for generalized multilinear forms. J. Multivariate Anal. 34 (1990), no. 2, 275–289.
  • V.H. de la Peña and E. Giné (1997). Decoupling. Springer-Verlag. Berlin Heidelberg New York.
  • Efron, B.; Stein, C. The jackknife estimate of variance. Ann. Statist. 9 (1981), no. 3, 586–596.
  • Godbole, Anant P. The exact and asymptotic distribution of overlapping success runs. Comm. Statist. Theory Methods 21 (1992), no. 4, 953–967.
  • Götze, F. On the rate of convergence in the multivariate CLT. Ann. Probab. 19 (1991), no. 2, 724–739.
  • Götze, F.; Tikhomirov, A. Asymptotic distribution of quadratic forms and applications. J. Theoret. Probab. 15 (2002), no. 2, 423–475.
  • Goldstein, Larry; Reinert, Gesine. Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 (1997), no. 4, 935–952.
  • Hájek, Jaroslav. Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist 39 1968 325–346.
  • Janson, Svante. Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge, 1997. x+340 pp. ISBN: 0-521-56128-0.
  • Karlin, Samuel; Rinott, Yosef. Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 10 (1982), no. 2, 485–501.
  • Kwapień, Stanisław; Woyczyński, Wojbor A. Random series and stochastic integrals: single and multiple. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xvi+360 pp. ISBN: 0-8176-3572-6.
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9.
  • Meyer, Paul-André. Quantum probability for probabilists. Lecture Notes in Mathematics, 1538. Springer-Verlag, Berlin, 1993. x+287 pp. ISBN: 3-540-56476-4.
  • Mossel, Elchanan; O'Donnell, Ryan; Oleszkiewicz, Krzysztof. Noise stability of functions with low influences: invariance and optimality. Ann. of Math. (2) 171 (2010), no. 1, 295–341.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37 (2009), no. 6, 2231–2261.
  • Nourdin, Ivan; Peccati, Giovanni; Réveillac, Anthony. Multivariate normal approximation using Stein's method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 45–58.
  • Nourdin, Ivan; Viens, Frederi G. Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14 (2009), no. 78, 2287–2309.
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5.
  • Nualart, David; Peccati, Giovanni. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005), no. 1, 177–193.
  • Peccati, G.; Solé, J. L.; Taqqu, M. S.; Utzet, F. Stein's method and normal approximation of Poisson functionals. Ann. Probab. 38 (2010), no. 2, 443–478.
  • Peccati, Giovanni; Taqqu, Murad S. Central limit theorems for double Poisson integrals. Bernoulli 14 (2008), no. 3, 791–821.
  • Peccati, Giovanni; Taqqu, Murad S. Wiener chaos: moments, cumulants and diagrams. A survey with computer implementation. Supplementary material available online. Bocconi & Springer Series, 1. Springer, Milan; Bocconi University Press, Milan, 2011. xiv+274 pp. ISBN: 978-88-470-1678-1.
  • Privault, Nicolas. Stochastic analysis of Bernoulli processes. Probab. Surv. 5 (2008), 435–483.
  • Privault, Nicolas; Schoutens, Wim. Discrete chaotic calculus and covariance identities. Stoch. Stoch. Rep. 72 (2002), no. 3-4, 289–315.
  • Reinert, Gesine. Three general approaches to Stein's method. An introduction to Stein's method, 183–221, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
  • Reinert, Gesine; Röllin, Adrian. Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 (2009), no. 6, 2150–2173.
  • Rinott, Yosef; Rotar, Vladimir. A multivariate CLT for local dependence with $n^ {-1/2}\log n$ rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 (1996), no. 2, 333–350.
  • Rinott, Yosef; Rotar, Vladimir. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics. Ann. Appl. Probab. 7 (1997), no. 4, 1080–1105.
  • Rotar', V. I. Limit theorems for multilinear forms and quasipolynomial functions. (Russian) Teor. Verojatnost. i Primenen. 20 (1975), no. 3, 527–546.
  • Rotar', V. I. Limit theorems for polylinear forms. J. Multivariate Anal. 9 (1979), no. 4, 511–530.
  • Stanley, Richard P. Enumerative combinatorics. Vol. 1. With a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 1997. xii+325 pp. ISBN: 0-521-55309-1; 0-521-66351-2.
  • Stein, Charles. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583–602. Univ. California Press, Berkeley, Calif., 1972.
  • Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes - Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0.
  • Surgailis, D. CLTs for polynomials of linear sequences: diagram formula with illustrations. Theory and applications of long-range dependence, 111–127, Birkhäuser Boston, Boston, MA, 2003.
  • van Zwet, W. R. A Berry-Esseen bound for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66 (1984), no. 3, 425–440.