Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Berry–Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences

Kai Krokowski, Anselm Reichenbachs, and Christoph Thäle

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Abstract

Berry–Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin–Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.

Résumé

Nous dérivons des estimations de type Berry–Esseen pour des fonctionnelles non-linéaires de suites infinies de Rademacher par la méthode de Malliavin–Stein. De plus, nous prouvons des extensions multivariées pour des vecteurs de fonctionnelles de Rademacher. Ces résultats établissent une connexion avec les probabilités de petites boules et apportent un nouvel éclairage sur la relation entre des théorèmes centraux limites sur le chaos de Rademacher et les normes d’opérateurs de contraction. Des applications concernent des succès consécutifs pondérés, un théorème central limite combinatoire et des traces de matrices de Bernoulli aléatoires.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 52, Number 2 (2016), 763-803.

Dates
Received: 3 April 2014
Revised: 1 October 2014
Accepted: 3 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367893

Digital Object Identifier
doi:10.1214/14-AIHP652

Mathematical Reviews number (MathSciNet)
MR3498009

Zentralblatt MATH identifier
1341.60005

Subjects
Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Berry–Esseen bound Central limit theorem Malliavin calculus Normal approximation Rademacher chaos Stein’s method

Citation

Krokowski, Kai; Reichenbachs, Anselm; Thäle, Christoph. Berry–Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 763--803. doi:10.1214/14-AIHP652. https://projecteuclid.org/euclid.aihp/1462367893.


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