THE FOURTH MOMENT THEOREM ON THE POISSON SPACE

• 成果类型: Article
• 署名作者: Dobler, Christian; Peccati, Giovanni
• 署名单位: University of Luxembourg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1215
• 发表日期: 2018
• 页码: 1878-1916
• 关键词: CENTRAL LIMIT-THEOREMS; U-statistics; gaussian fluctuations; normal approximation; gamma limits; CONVERGENCE; chaos

摘要

We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result-that has been elusive for several years-shows that the so-called 'fourth moment phenomenon', first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177-193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carre-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439-2459] and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341-2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.