Polarization Identity for Fourth Cumulants

• The paper introduces the unique polarization identity for fourth cumulants, extending the diagonal evaluation to a fully symmetric 4-linear form.
• It employs combinatorial sign-sum and inclusion–exclusion formulas to derive explicit moment-cumulant relationships, facilitating efficient computation.
• Applications span classical, quantum, and noncommutative settings, underpinning advanced limit theorems and higher-order dependency analysis.

The polarization identity for fourth cumulants provides the unique extension of the fourth moment polynomial along the diagonal—such as $C_4(x,x,x,x)$—to the fully symmetric 4-linear cumulant $C_4(x_1, x_2, x_3, x_4)$. This identity is fundamental to the theory and application of cumulants in commutative and noncommutative settings, underpins central limit theorems in chaos expansions, and enables efficient computation of higher-order dependencies in both classical and quantum statistical analysis.

1. Construction of the Polarization Identity for Fourth Cumulants

Let $\phi(h) = C_4(h, h, h, h)$ denote the homogeneous quartic polynomial corresponding to the univariate fourth cumulant of a random variable %%%%3%%%%, given explicitly by: $\phi(h) = \E[h^4] - 4\E[h^3]\E[h] - 3(\E[h^2])^2 + 12\E[h^2](\E[h])^2 - 6(\E[h])^4$ as shown in (Schefczik et al., 2019). Polarization extends $\phi$ to a symmetric 4-linear map, yielding the canonical multilinear fourth cumulant $C_4(x_1,x_2,x_3,x_4)$ by the formula: $$C_4(x_1, x_2, x_3, x_4) = \frac{1}{2^4 4!} \sum_{\varepsilon_i = \pm1} (\varepsilon_1 \varepsilon_2 \varepsilon_3 \varepsilon_4) \; \phi\left(\sum_{j=1}^4 \varepsilon_j x_j \right)$$ This "sign–sum" formula symmetrically aggregates diagonal evaluations over all sign combinations, ensuring that %%%%6%%%% is both multilinear and invariant under permutation of arguments (Schefczik et al., 2019, Thomas, 2013).

An equivalent "difference" or inclusion–exclusion form is: $$C_4(x_1,x_2,x_3,x_4) = \frac{1}{4!} \sum_{k=0}^4 (-1)^{4-k} \sum_{1 \le i_1 < \cdots < i_k \le 4} \phi(x_{i_1} + \cdots + x_{i_k})$$ This forms the basis for all further explicit cumulant expressions.

2. Raw-Moment Expansion of the Fourth Cumulant

Upon substituting the explicit polynomial expansion for $\phi$, the polarization sum yields the well-known raw-moment formula for $C_4$ (not assumed mean-zero): $\begin{aligned} C_4(x_1, x_2, x_3, x_4) =\;& \E[x_1 x_2 x_3 x_4] - \sum_{(\ell)} \E[x_{i} x_{j} x_{k}] \E[x_{\ell}] - \sum_{(ij)(k\ell)} \E[x_i x_j] \E[x_k x_\ell] \ &+ 2 \sum_{(ij)(k)(\ell)} \E[x_i x_j] \E[x_k] \E[x_\ell] - 6 \E[x_1] \E[x_2] \E[x_3] \E[x_4] \end{aligned}$ where permutation sums run over all tuples as detailed in (Schefczik et al., 2019). Exactly 19 terms survive, accounting for every symmetry in the argument indices. Centralizing variables (taking them mean-zero) further reduces the expansion.

3. Alternative Polarization Forms and Multivariate Interpretations

The same multilinearization process can be applied in more abstract settings. For any symmetric 4-linear map $u$, (Thomas, 2013) provides combinatorial and diagonal forms:

Formula Type	Expression	Source
"Sign-sum"	$$\displaystyle \frac{1}{384} \sum_{\epsilon_i = \pm 1} (\epsilon_1 \dots \epsilon_4) u(\epsilon_1 x_1 + \dots + \epsilon_4 x_4)$$	(Thomas, 2013)
"Inclusion–exclusion"	$$\displaystyle \frac{1}{24} \left[ u(\Sigma) - \sum_{ijk} u(x_i+x_j+x_k) + \sum_{ij} u(x_i+x_j) - \sum_i u(x_i) \right]$$	(Thomas, 2013)
Diagonal cumulant	$\displaystyle \kappa_4(x,x,x,x) = \E[x^4] - 3(\E[x^2])^2$ (for centered $x$)	(Thomas, 2013)

Specializing $u$ to the fourth cumulant functional produces the diagonal-moment formula: $\kappa_4(X_1, \dots, X_4) = \frac{1}{24}\left\{ \E\Big[(\textstyle\sum_i X_i)^4\Big] - \sum_{ijk} \E\Big[(\sum X_i)^4\Big] + \cdots \right\}$ which expresses all mixed cumulants in terms of diagonal fourth moments (Thomas, 2013).

4. Polarization for Fourth Cumulants in Noncommutative Chaos

In Wiener–Itô (classical), free Wigner, and $q$–Gaussian (noncommutative) chaos, polarization identities relate the fourth cumulant of sums to contractions of the summands. For random variables $X = I_m^W(f)$, $Y = I_n^W(g)$ in classical Wiener–Itô chaos, with $m$ and $n$ of opposite parity: $c_4(X + Y) = c_4(X) + c_4(Y) + 6 \, \Cov(X^2, Y^2)$ where $c_4(Z) = \E[Z^4] - 3 (\E[Z^2])^2$, and the covariance contracts the underlying symmetric kernels (Kemp et al., 25 Nov 2025). The parity condition ensures vanishing of $X^3 Y$ and similar cross-chaos interactions.

Analogous polarization formulas hold for free and $q$–Gaussian chaoses with covariance and mixed trace terms, each term corresponding to extensions of the original polarization principle. In all cases, these identities provide the algebraic backbone for higher-moment limit theorems and contraction-based characterization of sum statistics in chaos expansions (Kemp et al., 25 Nov 2025).

5. Polarization Identity in Quantum and Multipole Contexts

In the context of quantum optics and polarization, the fourth cumulant of an arbitrary projection of the Stokes operator is expressed via multipole moments: $$\kappa_{4}(\mathbf{n}) = Q_4(\mathbf{n}) - 3\ [Q_2(\mathbf{n})]^2$$ with $$Q_K(\mathbf{n}) = \sqrt{\frac{2K+1}{4\pi}} \sum_{q=-K}^K M_{Kq}\, Y_{Kq}(\theta_0, \varphi_0)$$ the $K$-th multipole component of the SU(2) $Q$-function, and $M_{Kq}$ the global polarization multipoles (Sanchez-Soto et al., 2013). Only the quadrupole $(K=2)$ and hexadecapole $(K=4)$ enter the fourth cumulant, sharply limiting the required tomography.

This multipole-based polarization identity is a direct analog of the combinatorial polarization in classical cumulants, confirming that, even in quantum observables, all higher-order dependencies are fully encoded in the $2^\text{nd}$ and $4^\text{th}$ spherical harmonics components.

6. Relation to Second-Order Polarization and Generalizations

The polarization identity for cumulants generalizes the bilinear identity for covariances. For $n=2$ and quadratic polynomial $Q(x) = C_2(x, x) = \E[x^2] - \E[x]^2$, polarization returns the familiar covariance: $C_2(x, y) = \frac{1}{2}[Q(x + y) - Q(x) - Q(y)] = \E[xy] - \E[x]\E[y]$ The quartic case is the unique symmetric $n=4$ extension: the same combinatorial structure as $n=2$ but in a higher-degree multilinear context. This generality holds for higher cumulants, though the combinatorics rapidly increase in complexity (Schefczik et al., 2019).

7. Applications, Structural Consequences, and Limitations

Polarization formulas for fourth cumulants are essential in unbiased estimation theory (Schefczik et al., 2019), limit theorems in stochastic integration and noncommutative probability (Kemp et al., 25 Nov 2025), and the paper of higher-order polarization in quantum states (Sanchez-Soto et al., 2013). The explicit moment-cumulant expressions facilitate algorithmic computation and theoretical analysis of mixed dependencies, especially under independence, orthogonality, or symmetry constraints.

A notable subtlety is the necessity of imposing parity or orthogonality constraints in noncommutative frameworks; without these, mixed cumulant terms survive and complicate the decomposition, as shown by the breakdown of polarization splits in $X + Y$ where $X$ and $Y$ live in identical chaos levels (Kemp et al., 25 Nov 2025). Thus, the practical use of these formulas requires careful assessment of the algebraic properties of the underlying variables.

In summary, the polarization identity for fourth cumulants is a central structural tool in multivariate statistical theory, with broad impact across both commutative and noncommutative probability, quantum state characterization, and higher-order statistical inference.