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Article Dans Une Revue Annals of Probability Année : 2016

Generalization of the Nualart-Peccati criterion

Résumé

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

Dates et versions

hal-01356783 , version 1 (26-08-2016)

Identifiants

Citer

Ehsan Azmoodeh, Dominique Malicet, Guillaume Mijoule, Guillaume Poly. Generalization of the Nualart-Peccati criterion. Annals of Probability, 2016, 44 (2), pp.924-954. ⟨10.1214/14-AOP992⟩. ⟨hal-01356783⟩
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