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arXiv · 0803.0458

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

Ivan Nourdin·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.

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Ivan Nourdin, Giovanni Peccati. 2009-12-09. Stein's method and exact Berry--Esseen asymptotics for functionals of Gaussian fields. https://doi.org/10.1214/09-aop461

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