Generalization of the Nualart–Peccati criterion

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 {Xn} n≥1 of multiple Wiener-Itô integrals of fixed order, if E[X

Quantitative clts on a gaussian space: a survey of recent developments

“…in [5,44,49,52]. The paper [24] provides similar estimates for random variables living inside the chaos of a Markov operator; further important developments can be found in [2,3]. Analogous statements in the setting of free probability are discussed in [13,22].…”

“…[5,44,49,52]); the second one deals with limit theorems for stochastic integrals in the framework of free probability (see e.g. [13,22]); the third one deals with fourth moment theorems for random variables living in the chaos of a general Markov diffusion generator (see [2,3,24]). The reader can consult the constantly updated webpage http://www.iecn.u-nancy.fr/∼nourdin/steinmalliavin.htm for many applications of Malliavin-type techniques, as well as for asymptotic results that are somehow connected with the fourth moment theorem proved in [42].…”

On a new Sheffer class of polynomials related to normal product distribution

“…Then, one of the main findings of [6] is that the polynomial family Q ≥0 characterizes the normal approximation on the Wiener chaoses in the sense that the following statements are equivalent: N (0, 1))] (= 0) for some P ∈ Q ≥0 .…”

“…Theorem 4.1 (Even moment Theorem [6]). Let {F n } n≥1 be a sequence of random elements in the fixed Wiener chaos of order p ≥ 2, and that IE[F 2 n ] = 1 for all n ≥ 1.…”

Stein operators for variables form the third and fourth Wiener chaoses

“…The case n = 2 corresponds to the centered chisquare random variable Z 2 − 1. This is a special case of the centered gamma distribution, for which the Malliavin-Stein method is also highly tractable; see [3,4,12,28]. As with Stein's method for normal approximation, at the heart of Stein's method for H 2 (Z) = Z 2 − 1 is the Stein equation…”