Coskewness: Third-Order Dependence Measure

• Coskewness is a third-order dependence measure quantifying asymmetric co-movement beyond covariance using formulations like centered mixed moments and standardized tensors.
• It encompasses various formulations such as L-coskewness, rank-based measures, and quantum three-mode correlators, highlighting its adaptability across statistical models.
• Establishing sharp extremal bounds and efficient computational methods for coskewness is crucial for its applications in risk management, portfolio optimization, and representation learning.

Coskewness is a third-order dependence measure, but the term is used in several non-equivalent ways across the recent literature. In a standardized centered mixed-moment framework, it is the first nontrivial higher-order centered mixed moment beyond covariance, given in the bivariate case by

$$\mu_2(X_1,X_2)=\mathbb{E}\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right].$$

In a trivariate standardized-moment framework it appears as

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$

while alternative formulations include $L$-coskewness, standardized rank coskewness, and normalized three-mode correlators in quantum optics (Bernard et al., 9 Aug 2025, Bernard et al., 2024, R et al., 14 Mar 2025, Minganti et al., 2022). Across these settings, coskewness quantifies asymmetric co-movement beyond covariance or correlation, and its attainable values depend on the marginals and the dependence structure rather than obeying a universal distribution-free range (Bernard et al., 2024).

1. Formal definitions and competing frameworks

The recent literature uses several formalizations of coskewness, each tailored to a different statistical object.

Framework	Formula	Scope
Centered mixed moment	$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$	Bivariate asymmetric dependence
Trivariate standardized moment	$$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$	Three-variable dependence
$L$-coskewness	$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$	Quantile-based bivariate comoment
Standardized rank coskewness	$$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$	Monotone-invariant trivariate dependence
Quantum three-mode coskewness	$$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$	Non-Gaussian three-mode correlations

In the centered mixed-moment hierarchy, the standardized centered mixed moment of order $d+1$ is

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$0

with $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$1 giving a Pearson correlation-type comoment, $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$2 coskewness, and $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$3 cokurtosis (Bernard et al., 9 Aug 2025). In this formulation, the means and standard deviations themselves do not matter for the standardized comoments; what matters is the dependence structure between the variables (Bernard et al., 9 Aug 2025).

The quantile-based literature replaces classical centered moments by $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$4-comoments. In the bivariate quantile-function model of “A quantile-based bivariate distribution,” coskewness is specifically $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$5-coskewness in the sense of Serfling and Xiao (2007), not the classical multivariate coskewness tensor (R et al., 14 Mar 2025). The dependence is embedded directly in the conditional quantile-density factor $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$6, so the coskewness is a derived functional of the distributional parameters rather than a primitive parameter (R et al., 14 Mar 2025).

In quantum optics, coskewness is used in a different but structurally analogous sense: a normalized third-order correlator of quadrature fluctuations. The paper on three-body ultrastrong coupling treats $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$7 as a witness of non-Gaussianity and of correlations that cannot be captured within semiclassical and Gaussian approximations (Minganti et al., 2022). This suggests that “coskewness” is best viewed as a family of third-order dependence descriptors whose exact normalization and interpretation depend on the ambient modeling framework.

2. Dependence uncertainty and sharp extremal bounds

A central recent theme is the behavior of coskewness under fixed marginals and unknown copula. In the actuarial setting, the sharp bounds for the raw mixed moment

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$8

are defined by

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$9

and Theorem 1 gives explicit sharp bounds together with the dependence structures that attain them (Bernard et al., 9 Aug 2025). For odd $L$0, the extremizers simplify to classical monotone couplings: the maximum is attained by the comonotonic coupling $L$1, and the minimum by the countermonotonic coupling $L$2 (Bernard et al., 9 Aug 2025). For even $L$3, the sign of $L$4 matters because $L$5; the optimizer is then a piecewise copula built from two inverse branches rather than a single monotone copula (Bernard et al., 9 Aug 2025).

The structural reason is that the distribution function of $L$6 is not one-to-one when $L$7 is even and $L$8 can take both positive and negative values. In that case, the optimal dependence structure becomes a mixture of two monotone branches. For symmetric, zero-mean $L$9, the inverse branches simplify to

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$0

and the maximizing and minimizing copulas become correspondingly explicit (Bernard et al., 9 Aug 2025).

The passage from raw to centered moments is direct. Proposition 4.1 states that bounding

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$1

is equivalent to bounding $$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$2 for standardized variables, so the same extremal dependence structures apply after standardization (Bernard et al., 9 Aug 2025). In this sense, coskewness inherits the full dependence-uncertainty geometry of the raw mixed moment.

For fully trivariate coskewness, the dependence-uncertainty problem is formulated through

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$3

and, for symmetric zero-mean marginals, the sharp bounds are

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$4

with attainment by explicit copulas; in odd dimension the optimizer is a cross product copula (Bernard et al., 2023). Specializing to $$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$5, the same machinery yields sharp coskewness bounds. For standardized symmetric marginals, explicit upper bounds reported in the literature include

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$6

with the lower bounds given by their negatives (Bernard et al., 2023).

These results make a precise distinction between marginal information and dependence information. In particular, they show that sharp coskewness statements require explicit control of the copula and, when only marginals are known, the relevant object is an extremal dependence problem rather than a single descriptive statistic.

3. Rank-based and quantile-based analogues

Several papers introduce monotone-invariant or quantile-based versions of coskewness. In the trivariate case, standardized rank coskewness is defined by

$$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$7

It is invariant under strictly increasing transformations, always lies in $$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$8, and equals $$\mu_2(X_1,X_2)=\mathbb{E}\!\left[\left(\frac{X_1-\mu_1}{\sigma_1}\right)\left(\frac{X_2-\mu_2}{\sigma_2}\right)^2\right]$$9 under independence (Bernard et al., 2023). The same paper positions $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$0 as a three-dimensional analogue of Spearman’s rho (Bernard et al., 2023).

A bivariate higher-order rank analogue appears in the actuarial literature as the standardized rank coefficient

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$1

for even $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$2, and

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$3

for odd $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$4. These coefficients satisfy $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$5, the bounds are sharp, and they are invariant under strictly increasing transformations (Bernard et al., 9 Aug 2025). In that setting, they serve as a rank-based, scale-free analogue of coskewness, similar in spirit to Spearman correlation (Bernard et al., 9 Aug 2025).

The quantile-based bivariate distribution literature uses $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$6-coskewness rather than standardized central moments. For $$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$7,

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$8

and, using the covariance representation of Cuadras (2002), the paper derives the quantile form

$$S(X_i,X_j,X_k)=\frac{\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)]}{\sigma_i\sigma_j\sigma_k}$$9

where $L$0 (R et al., 14 Mar 2025). In this model, positive $L$1-coskewness is interpreted as indicating that the variables undergo positive deviation at the same time, and the first cable-lifetime dataset yields the reported values

$L$2

(R et al., 14 Mar 2025).

These constructions separate two aims that are often conflated. Classical centered coskewness measures asymmetric dependence on the original scale, whereas rank-based and $L$3-moment formulations seek monotone invariance or quantile robustness. This suggests a division of labor: centered coskewness is natural when original-scale tail asymmetry matters, while rank and $L$4-comoment analogues are natural when dependence should be isolated from marginal shape.

4. Decoupling from correlation and the leakage problem

A recurrent misconception is that correlation and coskewness should be linked. The recent literature explicitly rejects that claim. “Modeling coskewness with zero correlation and correlation with zero coskewness” proves two separation results: one can have arbitrary admissible coskewness with zero pairwise correlations, and one can have arbitrary correlations with zero coskewness (Bernard et al., 2024). For symmetric marginals, the first construction uses mixtures of extremal coskewness copulas while maintaining

$L$5

and the second uses trivariate Gaussian vectors, for which

$L$6

for any admissible correlation structure (Bernard et al., 2024).

The same paper extends the separation to rank dependence. For arbitrary continuous strictly increasing marginals, standardized rank coskewness can range arbitrarily in $L$7 while all pairwise rank correlations vanish, and Gaussian copulas permit arbitrary pairwise rank correlations with

$L$8

(Bernard et al., 2024). It also gives a striking example: under comonotonicity,

$L$9

while all pairwise rank correlations are $$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$0 (Bernard et al., 2024). The conclusion is exact: there is no universal monotone or deterministic relationship between correlation and coskewness, or between rank correlation and standardized rank coskewness (Bernard et al., 2024).

A different but related issue arises in conditional maximum-entropy models of neural activity. For sparse correlated binary inputs, higher-order features can project onto first-order features through a coskewness leakage channel (Safaai et al., 1 Jun 2026). With

$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$1

the centered interaction $$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$2 projects onto a first-order coordinate through

$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$3

For binary inputs $$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$4, the identity

$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$5

implies

$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$6

which the paper identifies as the explicit coskewness form of leakage (Safaai et al., 1 Jun 2026). In sparse data, $$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$7, so $$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$8, making covariance almost directly convertible into a leakage channel (Safaai et al., 1 Jun 2026).

The corresponding information-geometric statement is

$$L_{3(i,j)}=\operatorname{Cov}\!\left(X_i,\ 6F_j(X_j)^2-6F_j(X_j)+1\right)$$9

so omitted higher-order, temporal, or hidden-state terms can shift fitted first-order parameters whenever the omitted and included sufficient statistics are correlated under the sampled input distribution (Safaai et al., 1 Jun 2026). The paper’s interpretive conclusion is that entropy explained by a direct MaxEnt fit is a prediction measure under the observed state distribution, not a mechanism-identification statistic (Safaai et al., 1 Jun 2026). In this setting, coskewness is not only a dependence descriptor but also the algebraic mechanism by which higher-order structure becomes statistically confounded with lower-order models.

5. Tensor structure, computational burden, and algorithmic surrogates

In multivariate settings, coskewness naturally becomes a third-order tensor. In principal skewness analysis for hyperspectral imagery, after centering and whitening,

$$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$0

is the supersymmetric coskewness tensor, and directional skewness is

$$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$1

The associated optimization leads to the tensor eigenpair equation

$$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$2

(Geng et al., 2019). The key technical point is that, unlike matrix eigenvectors, eigenvectors of a supersymmetric tensor are not inherently orthogonal in general (Geng et al., 2019). Nonorthogonal Principal Skewness Analysis therefore replaces the usual orthogonal deflation strategy by a Kronecker-structured projection, enlarging the feasible search space and reducing the improved update complexity from storage $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$3, complexity $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$4, to storage $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$5, complexity $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$6 (Geng et al., 2019).

The same tensor-growth problem appears in high-dimensional learning. In self-supervised action recognition, the moment-descriptor viewpoint treats mean, covariance, coskewness, and cokurtosis as first-, second-, third-, and fourth-order summaries, but the paper explicitly notes that the numbers of coefficients grow linearly, quadratically, cubically, and quartically with feature dimension (Wang et al., 2020). Its practical response is to keep mean, a low-rank covariance subspace, skewness, and kurtosis rather than full coskewness and cokurtosis tensors (Wang et al., 2020).

Large-scale higher-moment portfolio optimization exhibits the same bottleneck. In the sample-moment MVSK model, the coskewness tensor is

$$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$7

but explicit higher-order tensors require $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$8 storage for coskewness and $$RS(X_1,X_2,X_3)=32\,\mathbb{E}\!\left[\left(F_1(X_1)-\frac12\right)\left(F_2(X_2)-\frac12\right)\left(F_3(X_3)-\frac12\right)\right]$$9 construction cost (Wang et al., 28 Apr 2026). The Yau affine-normal descent method avoids explicit construction of $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$0 and $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$1 by working directly with the centered return matrix $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$2; the objective, gradient, Hessian-vector product, and third-order directional action can then each be evaluated in $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$3 arithmetic and $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$4 working memory (Wang et al., 28 Apr 2026).

A further response is regularization rather than explicit tensor estimation. In Discrete World Models via Regularization, coskewness is penalized through

$$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$5

restricted to distinct indices so that only genuine triplet interactions are penalized (Bizzaro et al., 2 Mar 2026). The paper’s rationale is explicit: pairwise decorrelation alone does not eliminate structured higher-order dependence, and minimizing $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$6 pushes latent bits toward joint factorization beyond second order (Bizzaro et al., 2 Mar 2026).

The simulation literature offers yet another compressed representation. In Random Orthogonal Matrix simulation, the full co-skewness tensor of standardized components is

$$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$7

assembled into $$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$8, while the Kollo skewness vector

$$\mathcal C_{abc}=\frac{\langle \hat x_a\hat x_b\hat x_c\rangle}{\sqrt{\langle \hat x_a^2\rangle\langle \hat x_b^2\rangle\langle \hat x_c^2\rangle}}$$9

is used as a lower-dimensional exact target for simulation (Alexander et al., 2020). This paper makes precise the tradeoff between preserving the full third-order object and targeting a structured summary that remains computationally manageable.

6. Applications in actuarial science, finance, quantum physics, and representation learning

In actuarial science, coskewness is treated as a material risk driver rather than a purely descriptive statistic. Under a copula-based mixture model with exponential marginals,

$d+1$0

so dependence moves continuously from the lower-extremal to the upper-extremal structure (Bernard et al., 9 Aug 2025). Within that model, expected shortfall

$d+1$1

increases monotonically with even-order centered mixed moments such as coskewness, whereas marginal expected shortfall

$d+1$2

is more nuanced, sometimes nearly unchanged and in general possibly non-monotone or flattening as $d+1$3 (Bernard et al., 9 Aug 2025). In life contingencies, last-survivor annuity values increase with dependence and coskewness, joint-life annuity values decrease, and the difference between minimal and maximal dependence grows with the annuity term $d+1$4 (Bernard et al., 9 Aug 2025). The practical implication is that mis-specifying higher-order dependence can induce significant annuity mispricing (Bernard et al., 9 Aug 2025).

In portfolio selection, coskewness enters as the third central sample moment of returns. The unrestricted MVSK objective

$d+1$5

rewards positive portfolio skewness through the $d+1$6 term, and the empirical study on a 5-minute A-share panel with $d+1$7 stocks finds that higher moments add the most value at moderate return targets (Wang et al., 28 Apr 2026). This indicates that coskewness can be economically useful, but not uniformly so across target regimes (Wang et al., 28 Apr 2026). A related option-implied object is the skew stickiness ratio

$d+1$8

which the rough-volatility literature treats as a normalized return–volatility covariation and interprets as a coskewness-like statistic; in Bergomi-type models its short-maturity limit is

$d+1$9

(Fukasawa, 5 Feb 2026).

In quantum optics, coskewness becomes a diagnostic of genuinely non-Gaussian criticality. The three-body ultrastrong-coupling Hamiltonian exhibits a first-order superradiant transition with broken $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$00 symmetry, and the ground-state coskewness $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$01 diverges near the critical point in a way not seen in ordinary two-body models (Minganti et al., 2022). The steady state under dissipation inherits the same qualitative signature (Minganti et al., 2022). Here coskewness functions as a witness that the critical state cannot be reduced to Gaussian fluctuations around a semiclassical displaced solution (Minganti et al., 2022).

In machine learning and representation learning, coskewness is used both positively and negatively: positively as a tensor or regularizer that captures third-order structure, and negatively as a complexity source to be compressed or avoided. The action-recognition literature treats full coskewness as theoretically informative but practically too large (Wang et al., 2020); the world-model literature penalizes it to suppress triplet entanglement (Bizzaro et al., 2 Mar 2026); and the neural identifiability literature shows that unmodeled coskewness can leak into first-order fits under sparse correlated sampling (Safaai et al., 1 Jun 2026). Across these domains, a common theme is that third-order dependence is neither negligible nor automatically interpretable from second-order summaries.

Taken together, these developments present coskewness as a structurally rich but context-dependent notion. It may denote a centered mixed moment, a rank-based dependence coefficient, an $$S(X_i,X_j,X_k)=\frac{\mathbb{E}\big[(X_i-\mu_i)(X_j-\mu_j)(X_k-\mu_k)\big]}{\sigma_i\sigma_j\sigma_k},$$02-comoment, a tensor of third-order interactions, or a normalized three-mode correlator. What unifies these uses is the attempt to quantify asymmetric dependence beyond covariance. What distinguishes them is the object being normalized, the invariances being sought, and the role played by dependence uncertainty, computation, and application-specific semantics.