On the fourth moment condition for Rademacher chaos

Abstract

Adapting the spectral viewpoint suggested in Ledoux (2012) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure by Döbler and Peccati (2017), we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e.\ non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order $m\ge 2$ with respect to a symmetric Rademacher sequence. This behavior, which is in contrast to the Gaussian (see Nualart and Peccati (2005)) and Poisson (see Döbler and Peccati (2017)) situation, closely resembles the conditions for asymptotic normality of degenerate, non-symmetric $U$-statistics from the classical paper by de Jong (1990).

arXiv comment40 pages; complete characterization of the symmetric case added; alternative proof extended to the general case; some mistakes fixed

References in corpus (6)

• Central limit theorems for sequences of multiple stochastic integrals
• Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
• Chaos of a Markov operator and the fourth moment condition
• Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals
• Multivariate central limit theorems for Rademacher functionals with applications
• The fourth moment theorem on the Poisson space

Cited by in corpus (2)

• Fourth moment theorems on the Poisson space in any dimension
• The fourth moment theorem on the Poisson space