INVARIANCE PRINCIPLES FOR HOMOGENEOUS SUMS: UNIVERSALITY OF GAUSSIAN WIENER CHAOS

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Peccati, Giovanni; Reinert, Gesine
• 署名单位: Universite Paris Cite; Sorbonne Universite; University of Luxembourg; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP531
• 发表日期: 2010
• 页码: 1947-1985
• 关键词: multiple stochastic integrals; central limit-theorems; steins method; normal approximation; malliavin calculus; multivariate clt; polylinear forms; functionals

摘要

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.