# Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields

Abstract : We show how to detect optimal Berry-Esséen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein's method and the method of moments and cumulants, and provide de facto local (one term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proved in Nourdin and Peccati (2009b). Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan (1994) and Ginovyan and Sahakyan (2007)), (ii) to exploding'' quadratic functionals of a Brownian sheet, and (iii) to a continuous-time version of the Breuer-Major CLT for functionals of a fractional Brownian motion.
Keywords :
Type de document :
Pré-publication, Document de travail
27 pages. To appear in The Annals of Probability. 2009
Domaine :

https://hal.archives-ouvertes.fr/hal-00260597
Contributeur : Ivan Nourdin <>
Soumis le : vendredi 30 janvier 2009 - 22:55:43
Dernière modification le : jeudi 16 mars 2017 - 01:07:39
Document(s) archivé(s) le : mercredi 22 septembre 2010 - 11:33:31

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Steinlocal-final-aop.pdf
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### Identifiants

• HAL Id : hal-00260597, version 2
• ARXIV : 0803.0458

### Citation

Ivan Nourdin, Giovanni Peccati. Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. 27 pages. To appear in The Annals of Probability. 2009. <hal-00260597v2>

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