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Proceedings of the American Mathematical Society

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The optimal fourth moment theorem


Authors: Ivan Nourdin and Giovanni Peccati
Journal: Proc. Amer. Math. Soc. 143 (2015), 3123-3133
MSC (2010): Primary 60F05, 60G15, 60H07
DOI: https://doi.org/10.1090/S0002-9939-2015-12417-3
Published electronically: March 18, 2015
MathSciNet review: 3336636
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Abstract: We compute the exact rates of convergence in total variation associated with the `fourth moment theorem' by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit theorem (CLT) if and only if the sequence of the corresponding fourth cumulants converges to zero. We also provide an explicit illustration based on the Breuer-Major CLT for Gaussian-subordinated random sequences.


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Additional Information

Ivan Nourdin
Affiliation: Université du Luxembourg. Faculté des Sciences, de la Technologie et de la Communication; UR en Mathématiques. 6, Rui Richard Coudenhove-Kalergi, L-1359 Luxembourg

Giovanni Peccati
Affiliation: Université du Luxembourg. Faculté des Sciences, de la Technologie et de la Communication; UR en Mathématiques. 6, Rui Richard Coudenhove-Kalergi, L-1359 Luxembourg

DOI: https://doi.org/10.1090/S0002-9939-2015-12417-3
Keywords: Berry-Esseen bounds, central limit theorem, fourth moment theorem, Gaussian fields, Stein's method, total variation
Received by editor(s): May 7, 2013
Received by editor(s) in revised form: October 11, 2013
Published electronically: March 18, 2015
Additional Notes: The first author was partially supported by the (French) ANR grant ‘Malliavin, Stein and Stochastic Equations with Irregular Coefficients’ [ANR-10-BLAN-0121]
The second author was partially supported by the grant F1R-MTH-PUL-12PAMP from the University of Luxembourg
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society

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