The Annals of Probability

Generalization of the Nualart–Peccati criterion

Ehsan Azmoodeh, Dominique Malicet, Guillaume Mijoule, and Guillaume Poly

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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,

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

Article information

Ann. Probab., Volume 44, Number 2 (2016), 924-954.

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

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 60H07: Stochastic calculus of variations and the Malliavin calculus 34L05: General spectral theory

Nualart–Peccati criterion Markov diffusive generators moment inequalities $\Gamma$-calculus Hermite polynomials spectral theory


Azmoodeh, Ehsan; Malicet, Dominique; Mijoule, Guillaume; Poly, Guillaume. Generalization of the Nualart–Peccati criterion. Ann. Probab. 44 (2016), no. 2, 924--954. doi:10.1214/14-AOP992.

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