Quantitative de Jong theorems in any dimension

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

“…where M F is defined in (8). Taking the supremum over the set of all 1-Lipschitz functions h : R m → R of class C 1 , we infer…”

“…A fourth moment theorem (FMT) is a mathematical statement, implying that a given sequence of centered and normalized random variables converges in distribution to a Gaussian limit, as soon as the corresponding sequence of fourth moments converges to 3 (that is, to the fourth moment of the standard Gaussian distribution). Distinguished examples of FMTs are, for example, de Jong's theorem for degenerate U -statistics (see [7,8]) as well as the CLTs for multiple Wiener-Itô integrals proved in [25,26]; the reader is referred to the webpage [12] for a list (composed of several hundreds of papers) of applications and extensions of such results, as well as to the lecture notes [31] for a modern discussion of their relevance in mathematical physics. Our first application of Theorem 1.2 is a quantitative multivariate fourth moment theorem for a vector of multiple Wiener-Itô integrals, considerably extending the qualitative multivariate results proved in [26].…”

The fourth moment theorem on the Poisson space

“…(b) We find it quite remarkable that our fourth moment bounds do not require any additional error term accounting for the discreteness of the Poisson space as is necessary, for example, in the context of degenerate U -statistics [see, e.g., de Jong (1990) and Döbler and Peccati (2017a)] as well as for discrete multiple integrals of independent Rademacher random variables [see Döbler and Krokowski (2017)] where such fourth moment theorems without remainder do not hold.…”

“…The so-called fourth moment phenomenon was first discovered in Nualart and Peccati (2005), where the authors proved that a sequence of normalized random variables, belonging to a fixed Wiener chaos of a Gaussian field, converge in distribution to a Gaussian random variable if and only if their fourth cumulant converges to zero. Such a result constitutes a dramatic simplification of the method of moments and cumulants [see, e.g., Nourdin and Peccati (2012), page 202], and represents a rough infinite-dimensional counterpart of classical results by de Jong; see de Jong (1987,1989,1990), as well as Döbler and Peccati (2017a), Döbler and Peccati (2017b) for recent advances. A particularly fruitful line of research was initiated in Nourdin and Peccati (2009a), where it is proved that the results of Nualart and Peccati (2005) can be recovered from very general estimates, obtained by combining the Malliavin calculus of variations with Stein's method for normal approximation.…”

Explicit rates of convergence in the multivariate CLT for nonlinear statistics

“…has been extensively discussed in the literature. The best known result given by de Jong [13] says that the σ −1 n W n converges to a standard normal random variable in distribution if It is interesting to mention that the condition (4.11) with i = j is equivalent to the fourth moment condition required in Theorem 1.7 of [14], provided that C ii = 1 for 1 ≤ i ≤ d.…”