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

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. © Institute of Mathematical Statistics, 2009.