| Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields (2008) | |||||||||||||
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| 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 (2007b). 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. | |||||||||||||
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