Generalization of the Nualart-Peccati criterion

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 \(\{X_n\}_{n\ge1}\) of multiple Wiener-It\^{o} integrals of fixed order, if \(\mathbb {E}[X_n^2]\to1\) and \(\mathbb...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2016-03
Main Authors: Azmoodeh, Ehsan, Malicet, Dominique, Mijoule, Guillaume, Poly, Guillaume
Format: Article
Language:English
Subjects:
Convergence
Criteria
Gaussian distribution
Integrals
Normal distribution
Random variables
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 \(\{X_n\}_{n\ge1}\) of multiple Wiener-It\^{o} integrals of fixed order, if \(\mathbb {E}[X_n^2]\to1\) and \(\mathbb {E}[X_n^4]\to \mathbb {E}[N^4]=3\). 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-It\^{o} integrals. For instance, if \(X\) is a normalized multiple Wiener-It\^{o} integral of order greater than one, \[\forall k\ge2,\qquad \mathbb {E}\bigl[X^{2k}\bigr]>\mathbb {E} \bigl[N^{2k}\bigr]=(2k-1)!!.\]
ISSN:2331-8422
DOI:10.48550/arxiv.1305.6579