Fourth-Moment Theorem for Weak Convergence

• The Fourth-Moment Theorem is a criterion establishing that convergence of the normalized fourth moment ensures weak convergence to Gaussian or semicircular laws.
• It simplifies the method of moments in complex settings by leveraging combinatorial, spectral, and Malliavin calculus techniques.
• Applications span Wiener, Wigner, and Poisson chaoses, random matrices, and infinite-dimensional frameworks with explicit quantitative convergence rates.

The fourth-moment theorem for weak convergence is a foundational result in modern probability theory, characterizing when sequences of structured random variables converge in distribution to Gaussian or semicircular laws based solely on the behavior of the fourth moment and minimal additional structural information. Originating in the paper of Wiener chaos and central limit theorems (CLTs), this phenomenon extends to Wigner chaos, q-Brownian, Poisson chaoses, functionals on random matrices, homogeneous sums, and, more recently, to infinite-dimensional frameworks such as Hilbert spaces. At its core, the fourth-moment theorem states that, under appropriate normalization and structural constraints, convergence of the (normalized) fourth moment to its Gaussian or semicircular value is both necessary and sufficient for weak convergence to the corresponding limit law. This result has profound implications: it drastically simplifies the method of moments in high-dimensional or noncommutative probability, provides sharp quantitative bounds, elucidates the universality phenomenon, and connects deeply with the spectral and combinatorial structure of the model.

1. Fundamental Statement and Scope of the Fourth-Moment Theorem

The archetypal fourth-moment theorem, due to Nualart and Peccati (2005), asserts that for a sequence of centered, unit-variance random variables $\{F_n\}$ in a fixed Wiener chaos of order $q \geq 2$, the condition

$\mathbb{E}[F_n^4] \to 3$

is equivalent to $F_n$ converging in distribution to the standard normal law. This result immediately generalizes to vector-valued settings (componentwise convergence of fourth moments), multiple integrals on Wigner chaos (semicircular law with fourth moment 2), and even q-Brownian chaoses (with limiting q-Gaussian laws and explicit combinatorial corrections to the fourth moment).

The theorem extends beyond chaos decompositions. For example, it governs the weak convergence of homogeneous sums (e.g., $$Q_X(f_n) = \sum f_n(i_1, \ldots, i_d) X_{i_1} \cdots X_{i_d}$$ for suitable kernels and independent inputs $X_i$), counts of monochromatic subgraphs under random colorings, and random matrix eigenvalues, provided that the relevant normalization and moment assumptions hold. In each setting, the fourth moment condition cleanly separates Gaussian (or semicircular) fluctuations from “non-Gaussian” contributions, with explicit moment correction terms reflecting the noncentral nature of the functional or the combinatorial complexity of the environment.

2. Necessity of the Fourth Moment: Random Matrix Theory

The necessity of the fourth moment is sharply demonstrated in random matrix theory, particularly for Hermitian Wigner matrices with independent, identically distributed (i.i.d.) upper-triangular entries of mean zero and variance one. The Four Moment Theorem ((Tao et al., 2010), Theorem 1.5) asserts that matching the first four moments of the atom distribution ensures that individual eigenvalue distributions agree to within $O(n^{-1/2-\epsilon})$ for large $n$, which is the scale of eigenvalue spacing in the bulk.

Conversely, if only the first three moments are matched, then the discrepancy in local eigenvalue statistics is of order $O(n^{-1/2+\epsilon})$. If two Wigner ensembles differ in their fourth moments, the mean locations of the eigenvalues shift by an amount $\sim n^{-1/2}$, as quantified by

$$\sum_{i=1}^n |\mathbb{E} \lambda_i(M_n) - \mathbb{E} \lambda_i(M_n')| \geq k n^{-1/2}$$

for some $k>0$ ((Tao et al., 2010), Theorem 1.6). This matches the typical level spacing and demonstrates that the fourth moment is not a technical artifact but rather essential for “local” universality and optimal localization of eigenvalues. The explicit asymptotics (Conjecture 1.7) show that the difference in fourth moments leads to deterministic shifts in mean eigenvalue locations at the scale of fluctuations, with the sensitivity encoded analytically by the Taylor expansion of high moments of the eigenvalues in terms of the matrix entry moments.

3. Combinatorial and Spectral Frameworks: Wiener, Wigner, and Poisson Chaos

At the technical heart of the fourth-moment phenomenon is the structure of chaos expansions and the way higher moments decompose via contractions and combinatorial partition lattices.

Gaussian Wiener Chaos

In Wiener chaos, the fourth moment theorem—proved via Malliavin calculus, Stein's method, and spectral decompositions—shows that, under normalization,

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

where $d_{TV}$ is total variation distance, and $C$ is a constant depending on the chaos order (Nourdin et al., 2013, Chen et al., 2014). If the third moment also vanishes (e.g., for polynomials of Hermite rank at least two), the bound improves optimally.

Wigner (Free) Chaos

For Wigner chaos, the analogous result (Kemp et al., 2010) is that convergence of the fourth moment to 2 ensures weak convergence to the semicircular law, with the crucial observation that all nontrivial contractions of the kernel must vanish. The proof deploys the combinatorics of noncrossing partitions and the corresponding free Malliavin calculus, enabling quantitative bounds in distances metrizing weak convergence (e.g., free analogues of Wasserstein distance).

q-Brownian and Poisson Chaos

In the q-Brownian context (Deya et al., 2012), the required fourth moment becomes $2 + q^{n^2}$ and, again, equivalence with weak convergence hinges on vanishing of nontrivial contractions. For Poisson chaos, the situation is more delicate: Gamma limits require the joint convergence of the third and fourth moments to their Gamma values (because the target law is non-symmetric), together with specific contraction norm conditions (Fissler et al., 2015). In all these cases, detailed combinatorial analysis of kernel contractions rigorously links the “non-Gaussian” part of the functional to explicit moment terms.

4. Quantitative and Universality Results

The fourth-moment theorem comes with quantitative convergence rates. In the Wiener chaos, the optimal rate in total variation is determined by the maximum of the third absolute moment and the fourth cumulant (Nourdin et al., 2013):

$$c \cdot M(F_n) \leq d_{TV}(F_n, N) \leq C \cdot M(F_n), \quad M(F_n) = \max\left\{|\mathbb{E}[F_n^3]|, |\mathbb{E}[F_n^4] - 3|\right\}$$

Universality results, such as those for homogeneous sums (Nourdin et al., 2014), show that the fourth-moment phenomenon is insensitive to the specific distribution of the underlying variables, provided minimal moment assumptions (e.g., variance one, vanishing third moment, and fourth moment exceeding a threshold) are met. In free probability, the threshold for the existence of the phenomenon is characterized precisely in terms of the excess fourth free cumulant.

A transfer principle connects convergence results from Poisson to Gaussian settings: if the fourth moment condition (properly normalized) is satisfied in the Poisson chaos, it transfers to the Gaussian chaos, even in multivariate situations (Döbler et al., 2017). This transfer is a key underpinning of universality for multilinear sums.

5. Extension to Infinite Divisibility, Diffusions, and Non-Gaussian Targets

The fourth-moment theorem generalizes to sequences of infinitely divisible laws. In classical, free, Boolean, or monotone convolution frameworks, convergence of the fourth moment to the appropriate value uniquely determines weak convergence to the corresponding limit law (Arizmendi, 2013, Arizmendi et al., 2014). This “one-moment” criterion extends to compound Poisson laws with finitely supported jumps, provided a finite number of moments match.

For diffusive generators beyond the Ornstein–Uhlenbeck operator, spectral analysis shows that weak convergence to Gaussian, Gamma, or Beta laws is again characterized via explicit moment conditions (including fourth moments) involving the carré du champ operator and spectral gaps (Azmoodeh et al., 2013). In the context of invariant measures of diffusions, a necessary and sufficient condition for total variation convergence can be written in terms of Malliavin derivatives and Stein factors involving the drift and diffusion coefficients (Kusuoka et al., 2013).

6. Extensions to Infinite-Dimensional Settings

Recent work extends the theorem to Hilbert-space valued random variables, particularly sequences of multiple Wiener–Itô integrals in infinite dimensions (Düker et al., 24 Sep 2025). Here, the main result states: if $F_n$ are $H$-valued multiple integrals (in fixed chaos), and the covariance operators $T_{F_n}$ converge to that of a nondegenerate Gaussian limit $T_Z$ in the trace-class norm,

$\|T_{F_n} - T_Z\|_S \to 0$

then $F_n$ converges in law to the Gaussian if and only if, for any orthonormal basis $\{e_j\}$,

$$\mathbb{E}[\langle F_n, e_j\rangle^4] \to \mathbb{E}[\langle Z, e_j\rangle^4]$$

for all $j$. The fourth moment convergence must hold in every direction (or, equivalently, for all continuous linear functionals). A Stein–Malliavin bound produces explicit quantitative rates for the distance metrizing weak convergence, expressed as

$$d_2(F, Z) \leq \frac{1}{2} \mathbb{E}\left[\| \langle D_M F, -D_M L^{-1}F \rangle_H - T_Z \|_S \right]$$

where $D_M$ denotes the Malliavin derivative and $L$ the Ornstein–Uhlenbeck generator.

7. Graph Combinatorics, Colorings, and Limit Theorems

The fourth-moment phenomenon manifests in combinatorial probability, such as counts of monochromatic subgraphs in random colorings (Bhattacharya et al., 2020, Das et al., 2022). For a normalized count $Z(H, G_n)$ of monochromatic copies of a graph $H$, weak convergence to normality is equivalent (for sufficiently large number of colors, e.g., $c \geq 30$) to $\mathbb{E}[Z(H, G_n)^4] \to 3$. The convergence rate is controlled by explicitly computable subgraph overlap counts (e.g., “good joins”):

$$\sup_x \left| P[Z(H, G_n) \leq x] - \Phi(x) \right| \lesssim \left( N(G, G_n) / N(Q, G_n)^2 \right)^{1/20}$$

where $N(G, G_n)$ counts certain error-dominant subgraph configurations, and $N(Q, G_n)$ counts “2-shared 2-joins.” The regimes in which the fourth moment both necessary and sufficient depend on critical combinatorial and coloring thresholds, which can be computed algorithmically from the moment and subgraph structure.

Conclusion

The fourth-moment theorem for weak convergence unifies and deeply influences the paper of stochastic fluctuations in structured probabilistic models. Its central insight—that appropriate, often minimal, moment conditions completely characterize Gaussian or semicircular limits—extends to a wide array of fields: random matrices, Wiener and Wigner chaoses, Poisson and q-chaos, homogeneous sums, infinite-dimensional functionals, and combinatorial graph-statistics. The theorem's reach reflects the universality of Gaussian phenomena in high-dimensional probability: asymptotically Gaussian behavior emerges robustly once the fourth moment aligns, provided model-specific combinatorial or algebraic constraints are satisfied. The theorem also provides a pathway to sharp quantitative bounds and motivates advances in Malliavin calculus, Stein’s method, spectral theory, and combinatorial probability.