Mathematics – Probability
Scientific paper
2008-03-04
Annals of Probability 2009, Vol. 37, No. 6, 2231-2261
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/09-AOP461 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
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.
Nourdin Ivan
Peccati Giovanni
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