GENERALIZATION OF THE NUALART-PECCATI CRITERION

• 成果类型: Article
• 署名作者: Azmoodeh, Ehsan; Malicet, Dominique; Mijoule, Guillaume; Poly, Guillaume
• 署名单位: University of Luxembourg; Universite Paris Saclay; Universite de Rennes
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP992
• 发表日期: 2016
• 页码: 924-954
• 关键词: CENTRAL LIMIT-THEOREMS; CONVERGENCE; functionals; chaos; zeros

摘要

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {X-n}(n >= 1) of multiple Wiener-Ito integrals of fixed order, if E[X-n(2)] -> 1 and E[X-n(4)] -> E[N-4] = 3. Since its appearance in 2005, the natural question of ascertaining which other moments can replace the fourth moment in the above criterion has remained entirely open. Based on the technique recently introduced in [J. Funct. Anal. 266 (2014) 2341-2359], we settle this problem and establish that the convergence of any even moment, greater than four, to the corresponding moment of the standard Gaussian distribution, guarantees the central convergence. As a by-product, we provide many new moment inequalities for multiple Wiener-Ito integrals. For instance, if X is a normalized multiple Wiener-Ito integral of order greater than one, for all k >= 2, E[X-2k] > E[N-2k] = (2k - 1)!!.