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March 2016 Generalization of the Nualart–Peccati criterion
Ehsan Azmoodeh, Dominique Malicet, Guillaume Mijoule, Guillaume Poly
Ann. Probab. 44(2): 924-954 (March 2016). DOI: 10.1214/14-AOP992


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–Itô 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–Itô integrals. For instance, if $X$ is a normalized multiple Wiener–Itô integral of order greater than one,

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


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Ehsan Azmoodeh. Dominique Malicet. Guillaume Mijoule. Guillaume Poly. "Generalization of the Nualart–Peccati criterion." Ann. Probab. 44 (2) 924 - 954, March 2016.


Received: 1 November 2013; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1342.60026
MathSciNet: MR3474463
Digital Object Identifier: 10.1214/14-AOP992

Primary: 33C45 , 34L05 , 47D07 , 60F05 , 60H07

Keywords: $\Gamma$-calculus , Hermite polynomials , Markov diffusive generators , Moment inequalities , Nualart–Peccati criterion , Spectral theory

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • March 2016
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