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

Ivan Nourdin (1) , Giovanni Peccati (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MODAL'X - Modélisation aléatoire de Paris X

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 (2009b). 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.

Mots clés: Berry-Esséen 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

Domaines: Probabilités [math.PR]

Ivan Nourdin, Giovanni Peccati. Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. 2009. ⟨hal-00260597v2⟩