STEIN'S METHOD AND EXACT BERRY-ESSEEN ASYMPTOTICS FOR FUNCTIONALS OF GAUSSIAN FIELDS

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Peccati, Giovanni
• 署名单位: Sorbonne Universite; Universite Paris Cite; Universite Paris Nanterre; Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP461
• 发表日期: 2009
• 页码: 2231-2261
• 关键词: CENTRAL LIMIT-THEOREMS; quadratic functionals; STOCHASTIC INTEGRALS; random-variables; forms; approximations; CONVERGENCE

摘要

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.