Four Moment Theorems in Probability

• Four Moment Theorems are convergence criteria in probability that require matching the first four moments to ensure convergence in law to a target distribution.
• They utilize techniques like Malliavin calculus and Stein’s method to establish quantitative error bounds across Wiener, Poisson, and free chaos settings.
• Applications include random matrix theory and combinatorial structures, underpinning universality results and minimal moment conditions in high-dimensional stochastic analysis.

Four Moment Theorems are a class of results in probability theory and stochastic analysis asserting that, for a wide array of random objects, convergence of the first four moments (or, in many contexts, specifically the second and fourth moments given centering and variance normalization) to those of a target law (often Gaussian, semicircular, or another Pearson family member) suffices to guarantee convergence in law. Originating in the paper of Wiener chaoses by Nualart and Peccati, these theorems have now been extended to settings including random matrix theory, Poisson and free chaos, Markov diffusions, and combinatorial structures, and they play an essential role in quantifying universality and the minimal moment conditions for asymptotic normality.

1. Foundations and Historical Origins

The prototypical fourth moment theorem was established by Nualart and Peccati (2005, see (Nourdin et al., 2013, Chen et al., 2014)), who proved that for a sequence $(F_n)$ of normalized random variables living in a fixed Wiener chaos of order $q \geq 2$, the convergence $E[F_n^4]\to 3$ (the fourth moment of the standard normal) is necessary and sufficient for $F_n$ to converge in law to the standard Gaussian. This result provided a sharp and surprisingly minimal moment criterion, bypassing the need to check higher moments or Lindeberg-type conditions, and using tools such as Malliavin calculus and Stein's method to derive explicit quantitative error bounds,

$d_{TV}(F_n, N) \leq C \sqrt{|E[F_n^4] - 3|},$

where $N \sim \mathcal{N}(0,1)$ and $C$ depends on the chaos order (Nourdin et al., 2013).

This phenomenon was subsequently generalized to Poisson chaoses, free probability (Wigner chaos), Markov diffusions (Laguerre, Jacobi), and combinatorial contexts, resulting in a flexible toolkit for proving universality results across probability theory (Azmoodeh et al., 2013, Döbler et al., 2017, Cébron, 2018, Tao et al., 2011).

2. Scope and Contexts of Application

Wiener and Gaussian Chaos

The fourth moment theorem for Wiener chaos asserts that for sequences $F_n = I_q(f_n)$ of multiple Wiener–Itô integrals of fixed order $q$ with $E[F_n^2]=1$,

$$F_n \xrightarrow{\text{law}} N(0,1) \iff E[F_n^4] \to 3.$$

Quantitative rates in total variation, Wasserstein, and other metrics are expressed directly in terms of the fourth cumulant (Nourdin et al., 2013, Chen et al., 2014).

Sums of Multiple Integrals and Infinite Chaos Expansions

Recent generalizations address random variables $X_n = Y_n + \mathcal{E}_n$ where $Y_n$ and $\mathcal{E}_n$ lie in different chaos orders, particularly when the orders have different parities. Under suitable independence or regularity with respect to the Ornstein–Uhlenbeck operator, a fourth-moment theorem persists; that is,

$$E[X_n^4] \to 3 \implies X_n \xrightarrow{\text{law}} N(0,1)$$

(Basse-O'Connor et al., 5 Feb 2025). Importantly, combinations of multiple integrals from different chaoses cannot themselves be exactly Gaussian under non-degenerate conditions.

Poisson Chaoses and Markov Diffusions

On the Poisson space, multiple Wiener–Itô integrals with respect to a compensated Poisson measure exhibit a similar fourth moment phenomenon (Döbler et al., 2017, Döbler et al., 2018), though analytic details involve contraction kernels and carré du champ operators:

• $E[F_n^4] \to 3$
• Vanishing of certain contraction norms,
• Variance of $\Gamma(F_n, F_n)$ converging to zero.

All are essentially equivalent for asymptotic normality, provided uniform integrability.

The Markov diffusion framework further extends the setting to general Pearson diffusions and related polynomial families, where convergence of the first four moments (precisely, certain linear combinations as determined by the generator specification) guarantees convergence to the corresponding Pearson distribution, illustrated via spectral and carré du champ arguments (Bourguin et al., 2018, Azmoodeh et al., 2013).

Free Probability (Wigner and Poisson Chaos)

In free probability, analogous theorems are established for homogeneous Wigner chaos (semicircular law as the limiting object) (Cébron, 2018, Bourguin et al., 2017), as well as for multidimensional free Poisson limits (Gao et al., 2017). The fourth moment condition, $\tau(F_n^4)\to 2$, suffices for convergence in law to the standard semicircular element. Quantitative bounds in terms of free Stein discrepancies and free Wasserstein distances have been derived.

Random Matrix Theory

In random matrix theory, the Four Moment Theorem (as in (Tao et al., 2011)) shows that matching the first four moments (up to order four off-diagonal; up to order two diagonal) of entries forces local eigenvalue statistics of Wigner matrices to coincide at the universality level. This principle underlies many universality results for eigenvalue distributions and spacing.

Combinatorics and Graph Coloring

Fourth moment theorems also emerge in combinatorial contexts, such as the normal approximation for the number of monochromatic triangles in random graph colorings. For $c\geq 5$ colors, matching the normalized fourth moment is both necessary and sufficient for a central limit theorem to hold (Bhattacharya et al., 2020).

3. Key Analytical Tools and Techniques

Malliavin Calculus and Stein’s Method

The proofs and extensions of four moment theorems draw on Malliavin calculus (via derivative operators and chaos decompositions) and Stein’s method (operator-based characterizations of the limiting law). In the Gaussian and Poisson settings, integration by parts formulae lead to variance identities for the carré du champ operator, or bounds involving the norms of Malliavin derivatives (Chen et al., 2014, Nourdin et al., 2013, Azmoodeh et al., 2013, Döbler et al., 2018).

Product and Contraction Formulas

A central role is played by product formulas for multiple integrals,

$$I_p(f) I_q(g) = \sum_{r=0}^{p\wedge q} r! {p \choose r}{q \choose r} I_{p+q-2r}(f \,\widetilde{\otimes}_r\, g)$$

and their analogues in Poisson and free settings, leading to explicit expressions for mixed moments and contraction norm conditions.

Quantitative Berry–Esséen Type Bounds

Optimal and explicit rates for convergence in various distances (total variation, Wasserstein, and related metrics) have been developed, with the bound's order determined by the square root (or a power) of the deviation of the fourth moment from its limiting value (Nourdin et al., 2013, Campese, 2015, Cébron, 2018, Chen et al., 2023).

4. Extensions: Complex, Multivariate, and Non-Gaussian Settings

Complex Chaos

Recent work generalizes four moment theorems to complex multiple Wiener–Itô integrals, establishing that asymptotic normality for sequences of ($p$, $q$)-order complex chaoses is determined by vanishing contraction norms involving reverse conjugate kernels (Chen et al., 2023, Campese, 2015). In multivariate cases, Berry–Esséen bounds exhibit dependence on partial order relations between chaos indices—a phenomenon with no real analogue.

Beyond Gaussian Targets

The fourth moment paradigm is extended to chaos eigenfunctions of diffusion generators targeting Beta, Gamma, or Pearson distributions; the precise moment combination guaranteeing convergence depends on the polynomial coefficients of the target diffusion's generator (Azmoodeh et al., 2013, Bourguin et al., 2018). In free and classical infinite divisibility settings, convergence of the fourth moment suffices for convergence to the appropriate Gaussian, semicircular, or arcsine laws (Arizmendi, 2013).

Universality and Transfer Principles

Universality principles demonstrate that once a fourth moment theorem is established for a model object (e.g., Gaussian or semicircular), analogous results carry over to broad classes of distributions or random structures with matching lower moments (Nourdin et al., 2014, Tao et al., 2011, Döbler et al., 2017).

5. Higher-Order Phenomena and Limitations

While fourth moment theorems provide sharp minimal criteria for many settings, there are environments (e.g., with heavy-tailed targets, certain low-color combinatorial statistics) where matching four moments is not sufficient, reflecting more subtle dependence on combinatorial or algebraic structure (Bhattacharya et al., 2020, Bourguin et al., 2018). The precise boundaries of the paradigm are an active area of research.

6. Representative Formulas and Paradigms

Context	Fourth Moment Criterion	Target Law
Wiener chaos ($q\geq 2$)	$E[F_n^4]\to 3$	Normal
Poisson chaos	$E[F_n^4]\to 3$	Normal
Free Wigner chaos	$\tau(F_n^4)\to 2$	Semicircular
Pearson/Markov chaos	$a E[X_n^4] + b E[X_n^3] + \dots \to C$	Pearson
Random matrices (Wigner)	Matching to 4th off-diagonal, 2nd diagonal moments	Universality

Central analytical objects:

• Contraction norms: $||f_n \otimes_r f_n|| \to 0$
• Carré du champ variance: $\mathrm{Var}(\Gamma(F_n,F_n))\to 0$
• Free Stein discrepancy: controls free Wasserstein distance via fourth moment (Cébron, 2018)
• Berry–Esséen bound: $d_W(F_n, N) \leq C \sqrt{|E[F_n^4] - m_4|}$

7. Synthesis and Impact

Four Moment Theorems provide an efficient, sharp, and powerful criterion for normal (and more generally, Pearson family) approximation in high-dimensional and non-linear stochastic analysis. Their impact is seen in:

• Simplification of CLT proofs for functionals of Gaussian, Poisson, and free fields.
• Rigorous universality results in random matrix theory.
• Quantitative limit theorems with explicit rates, integral to applications in statistical mechanics, stochastic geometry, combinatorial probability, and more.
• Extensions to complex-valued functionals, Markov operator theory, and beyond.

They unify probabilistic, analytic, and combinatorial perspectives on convergence in law, leveraging tools from chaos expansions, Malliavin calculus, Stein’s method, spectral analysis, and algebraic combinatorics. Future research avenues include refining moment conditions in new settings, exploring multidimensional phenomena, and identifying necessary versus sufficient regimes beyond the classical universality domains.