Generalization of the Nualart-Peccati criterion

TL;DR

This paper extends the Nualart-Peccati criterion by proving that convergence of any even moment greater than four to the Gaussian moment guarantees distributional convergence for multiple Wiener-Itô integrals, broadening the criterion's applicability.

Contribution

It establishes that any even moment above four converging to the Gaussian moment ensures convergence in distribution, generalizing the original fourth-moment condition.

Findings

Convergence of any higher even moment implies Gaussian convergence.

New moment inequalities for multiple Wiener-Itô integrals.

Generalization of the Nualart-Peccati criterion to all even moments above four.

Abstract

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, \[\forall k\ge2,\qquad \mathbb {E}\bigl[X^{2k}\bigr]>\mathbb {E} \bigl[N^{2k}\bigr]=(2k-1)!!.\]