The Annals of Probability

Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

Ivan Nourdin and Giovanni Peccati

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Abstract

We show how to detect optimal Berry–Esseen 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 proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (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.

Article information

Source
Ann. Probab., Volume 37, Number 6 (2009), 2231-2261.

Dates
First available in Project Euclid: 16 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1258380788

Digital Object Identifier
doi:10.1214/09-AOP461

Mathematical Reviews number (MathSciNet)
MR2573557

Zentralblatt MATH identifier
1196.60034

Subjects
Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Berry–Esseen bounds Breuer–Major CLT Brownian sheet fractional Brownian motion local Edgeworth expansions Malliavin calculus multiple stochastic integrals normal approximation optimal rates quadratic functionals Stein’s method Toeplitz quadratic forms

Citation

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. doi:10.1214/09-AOP461. https://projecteuclid.org/euclid.aop/1258380788


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