[Paper Review] On the fourth moment condition for Rademacher chaos

Christian Döbler, Kai Krokowski|arXiv (Cornell University)|Jun 2, 2017

Stochastic processes and statistical mechanics21 references3 citations

TL;DR

This paper establishes that the fourth moment condition alone is insufficient to guarantee asymptotic normality for discrete multiple integrals (chaos) of order $m \geq 2$ with respect to general, non-symmetric and non-homogeneous Rademacher sequences. Instead, the maximal influence of the kernel functions—quantified via $\sup_k \operatorname{Inf}_k(f_n)$—must also vanish in the limit. The key contribution is a refined fourth moment theorem that combines convergence of the fourth moment to 3 and vanishing maximal influence, which ensures convergence to a normal distribution, contrasting sharply with the classical Gaussian and Poisson settings where the fourth moment alone suffices.

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 $m\geq2$ 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).

Motivation & Objective

• To investigate whether the fourth moment condition alone ensures asymptotic normality for discrete multiple integrals of order $m \geq 2$ with respect to general Rademacher sequences.
• To determine whether the classical fourth moment theorem—valid in Gaussian and Poisson settings—extends to the Rademacher chaos framework.
• To identify additional conditions beyond the fourth moment that are necessary for asymptotic normality in the Rademacher setting.
• To establish a quantitative normal approximation bound using the Malliavin-Stein method in the discrete, non-symmetric Rademacher setting.

Proposed method

• Adopt a spectral viewpoint via the carré du champ operator, inspired by Ledoux (2012) and extended to Poisson processes in [DP17b], to analyze the structure of Rademacher chaos.
• Define the influence of variable $k$ on a kernel $f$ as $\operatorname{Inf}_k(f) = \sum_{i_2 < \cdots < i_m} f^2(k, i_2, \dots, i_m)$, capturing the sensitivity of the functional to individual Rademacher variables.
• Use discrete Malliavin calculus and Stein's method to derive a quantitative bound on the Wasserstein distance between a normalized Rademacher chaos $F_n$ and a standard normal variable.
• Establish a bound of the form $d_{\mathcal{W}}(W_n, N) \leq C_1 \sqrt{|\mathbb{E}[W_n^4] - 3|} + C_2 \varrho(W_n)$, where $\varrho(W_n)$ measures the maximal influence of the kernel.
• Prove that if $\mathbb{E}[F_n^4] \to 3$ and $\sup_k \operatorname{Inf}_k(f_n) \to 0$ as $n \to \infty$, then $F_n \xrightarrow{d} N(0,1)$.
• Demonstrate that the fourth moment condition alone is insufficient for asymptotic normality in the Rademacher chaos, unlike in the Gaussian or Poisson cases.

Experimental results

Research questions

• RQ1 Does the fourth moment condition $\mathbb{E}[F_n^4] \to 3$ alone imply asymptotic normality for discrete multiple integrals of order $m \geq 2$ with respect to a general Rademacher sequence?
• RQ2 What additional conditions, beyond the fourth moment, are required to ensure asymptotic normality in the Rademacher chaos setting?
• RQ3 How does the maximal influence $\sup_k \operatorname{Inf}_k(f_n)$ of the kernel functions affect the convergence to normality in the Rademacher chaos?
• RQ4 Can a quantitative normal approximation bound be derived in the Rademacher chaos framework using the Malliavin-Stein method, and how does it depend on the fourth moment and influence structure?
• RQ5 To what extent does the behavior of Rademacher chaos resemble that of degenerate, non-symmetric $U$-statistics, as studied in [dJ90]?

Key findings

• The fourth moment condition $\mathbb{E}[F_n^4] \to 3$ is not sufficient for asymptotic normality in the Rademacher chaos of order $m \geq 2$ when the Rademacher sequence is non-symmetric or non-homogeneous.
• Asymptotic normality holds if and only if both $\mathbb{E}[F_n^4] \to 3$ and $\sup_k \operatorname{Inf}_k(f_n) \to 0$ as $n \to \infty$, showing that influence control is essential.
• The paper establishes a quantitative Wasserstein bound: $d_{\mathcal{W}}(W_n, N) \leq C_1 \sqrt{|\mathbb{E}[W_n^4] - 3|} + C_2 \varrho(W_n)$, where $\varrho(W_n) = \sup_k \operatorname{Inf}_k(f_n)$, confirming the necessity of influence decay.
• The result reveals a fundamental difference from the Gaussian and Poisson settings, where the fourth moment alone suffices for asymptotic normality.
• The behavior closely mirrors that of degenerate, non-symmetric $U$-statistics from [dJ90], where influence control is also required.
• The authors prove that there is no exact fourth moment theorem for discrete multiple integrals of order $m \geq 2$ with respect to a symmetric Rademacher sequence, due to the failure of the fourth moment to control influence.