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November 2009 Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields
Ivan Nourdin, Giovanni Peccati
Ann. Probab. 37(6): 2231-2261 (November 2009). DOI: 10.1214/09-AOP461


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.


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Ivan Nourdin. Giovanni Peccati. "Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields." Ann. Probab. 37 (6) 2231 - 2261, November 2009.


Published: November 2009
First available in Project Euclid: 16 November 2009

zbMATH: 1196.60034
MathSciNet: MR2573557
Digital Object Identifier: 10.1214/09-AOP461

Primary: 60F05 , 60G15 , 60H05 , 60H07

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

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 6 • November 2009
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