Quantifying Dependence Between Random Vectors: A New Index with Applications
Pith reviewed 2026-05-19 18:44 UTC · model grok-4.3
The pith
A new index for random vectors equals zero exactly when they are sub-independent and takes all values in [0,1].
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper proposes a dependence index for random vectors that is constructed from their characteristic functions, normalized to lie in the interval [0,1], and satisfies the property that the index equals zero if and only if the vectors are sub-independent. The index possesses an alternative representation in terms of moments of the component variables. A corresponding empirical estimator is derived that admits an efficient computational formula, and the asymptotic distribution of this estimator is established under standard regularity conditions.
What carries the argument
The dependence index obtained by normalizing an integral or expectation involving the difference between the joint characteristic function and the product of the marginal characteristic functions.
If this is right
- The index supplies a direct numerical test for the presence or absence of sub-independence in observed data.
- Its moment-based form permits evaluation without repeated numerical integration of characteristic functions.
- The derived asymptotic theory for the estimator supports confidence intervals and hypothesis tests in large samples.
- The same construction can be applied to quantify dependence in machine-learning feature sets, in joint risk models in actuarial science, and in dependent inter-arrival times in renewal processes.
Where Pith is reading between the lines
- The index could be extended to three or more random vectors by replacing the pairwise characteristic-function product with the appropriate marginal product.
- In high-dimensional feature selection the index might serve as a penalty term that discourages both complete independence and strong linear dependence.
- Comparisons with distance correlation or other kernel-based measures could reveal which dependence notions each index is most sensitive to.
- The moment representation may allow closed-form expressions for the index under common parametric families such as multivariate normals or copula models.
Load-bearing premise
The specific formula built from characteristic functions is assumed to return exactly zero under sub-independence and a positive number otherwise.
What would settle it
Compute the index on a concrete pair of sub-independent but non-independent random vectors (for example, two suitably scaled non-Gaussian variables with zero covariance) and check whether the numerical value is exactly zero; repeat the calculation on fully independent vectors to confirm the value is zero and on dependent vectors to confirm the value is positive.
read the original abstract
This article proposes a new index for quantifying the degree of dependence between random vectors. The index takes values in [0,1] and equals zero if and only if the random vectors are sub-independent. Unlike mere uncorrelatedness, sub-independence implies a stronger form of dependence while remaining strictly weaker than full independence. The proposed index is constructed via characteristic functions and admits a simplified representation in terms of moments. We establish its theoretical properties and derive a computationally efficient formula for the corresponding empirical measure. Furthermore, we investigate the asymptotic behavior of the estimator and demonstrate its practical utility through applications in machine learning, actuarial science, and renewal theory.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a new index for quantifying dependence between random vectors, constructed via characteristic functions. The index is claimed to take values in [0,1] and to equal zero if and only if the vectors are sub-independent (a relation strictly stronger than uncorrelatedness but weaker than independence). It admits a simplified moment-based representation, with theoretical properties established, a computationally efficient empirical estimator derived, asymptotic behavior of the estimator analyzed, and applications illustrated in machine learning, actuarial science, and renewal theory.
Significance. If the central claims hold, this work is significant for introducing a dependence measure targeted at sub-independence, filling a gap between correlation and full independence. The characteristic-function construction, its reduction to moments, and the rigorous establishment of the [0,1] range together with the iff property (via integral representations that vanish precisely under appropriate factorization) are clear strengths. The development of an empirical formula with asymptotic guarantees further supports practical deployment. Credit is due for the absence of hidden gaps in the integrability and positivity arguments.
minor comments (3)
- Abstract: the term 'sub-independent' is introduced without a brief definition or reference; adding one sentence would improve accessibility for readers outside the immediate subfield.
- Empirical measure section: while a computationally efficient formula is stated, an explicit pseudocode or step-by-step algorithm for its implementation would enhance reproducibility.
- Applications: the demonstrations in machine learning, actuarial science, and renewal theory are outlined at a high level; more quantitative comparison against existing dependence measures (e.g., distance correlation) would strengthen the practical-utility claim.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, including the recognition of the index's construction via characteristic functions, its [0,1] range, the iff property for sub-independence, the moment-based representation, the empirical estimator, and the applications. The recommendation for minor revision is appreciated.
Circularity Check
No significant circularity; derivation self-contained
full rationale
The index is defined directly from an integral involving the joint and marginal characteristic functions. The proof that it equals zero precisely under sub-independence follows from the integral vanishing if and only if the joint characteristic function factors in the required manner, using only standard analytic properties of characteristic functions. The [0,1] bounds, moment simplification, and estimator asymptotics are likewise derived from the same integral representation without invoking fitted parameters, self-referential definitions, or load-bearing self-citations. All steps remain independent of the target claims.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Characteristic functions characterize the joint distribution sufficiently to detect sub-independence.
invented entities (1)
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New dependence index
no independent evidence
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
siCov(α)(X1,…,Xn) = ∫ |ϕ_{X1,…,Xn}(t,…,t) − ∏ ϕ_{Xi}(t)|^2 ρα(t) dt with ρα(t) = c(p,α)^{-1}|t|^{-α-p}
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
-
[1]
Statistical Theory of Reliability and Life Testing: Probability Models
Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. Hot, Rinehart and Winston, Inc, New 28 York
work page 1975
-
[2]
Convexity and measures of statistical association
Borgonovo, E., Figalli, A., Ghosal, P., Plischke, E., Savar´ e, G., 2025. Convexity and measures of statistical association. J. R. Stat. Soc. Ser. B Stat. Method. 87, 1281-1304
work page 2025
- [3]
-
[4]
Estimation of varying coefficient models with measurement error
Dong, H., Otsu, T., Taylor, L., 2022. Estimation of varying coefficient models with measurement error. J. Econometrics 230(2), 388-415
work page 2022
- [5]
-
[6]
Ebrahimi, N., Hamedani, G. G., Soofi, E. S., Volkmer, H., 2010. A class of models for uncorrelated random variables. J. Multivariate Anal. 101(8), 1859-1871
work page 2010
-
[7]
On relationships between the Pearson and the distance correlation coefficients
Edelmann, D., Mori, T.F., Szekely, G.J., 2021. On relationships between the Pearson and the distance correlation coefficients. Stat. Probab. Lett. 169, 108960
work page 2021
-
[8]
The distance standard deviation
Edelmann, D., Richards, D., Vogel, D., 2020. The distance standard deviation. Ann. Statist. 48, 3395-3416
work page 2020
-
[9]
Hamedani, G.G., Maadooliat, M., 2015. Sub-independence: A Useful Concept (Mathematics Research Developments), Nova Science Publish- ers, New York
work page 2015
-
[10]
Probability inequalities for sums of bounded ran- dom variables
Hoeffding, W., 1963. Probability inequalities for sums of bounded ran- dom variables. J. Amer. Statist. Assoc. 58(301), 13-30. 29
work page 1963
-
[11]
Modern Actu- arial Risk Theory: Using R (2nd ed.)
Kaas, R., Goovaerts, M., Dhaene, J., Denuit, D., 2008. Modern Actu- arial Risk Theory: Using R (2nd ed.). Springer (Springer-Verlag Berlin Heidelberg)
work page 2008
-
[12]
Distance correlation test for high- dimensional independence
Li, W., Wang, Q., Yao, J., 2024. Distance correlation test for high- dimensional independence. Bernoulli 30(4), 3165-3192
work page 2024
-
[13]
Differential distance correlation and its applications
Liu, Y., Pengjian Shang, P., 2026. Differential distance correlation and its applications. J. Multivariate Anal. 214, 105631
work page 2026
-
[14]
Mitov, K. V., Omey, E., 2014. Renewal Processes, SpringerBriefs in Statistics, Springer International Publishing
work page 2014
-
[15]
Foundations of Ma- chine Learning
Mohri, M., Rostamizadeh, A., Talwalkar, A., 2018. Foundations of Ma- chine Learning. MA: MIT Press (2nd ed), Cambridge
work page 2018
-
[16]
Schennach, S. M. 2019. Convolution without independence. J. Econo- metrics 211(1), 308-318
work page 2019
- [17]
-
[18]
Martingale difference correlation and its use in high-dimensional variable screening
Shao, X., Zhang, J., 2014. Martingale difference correlation and its use in high-dimensional variable screening. J. Amer. Statist. Assoc. 109, 1302-1318
work page 2014
-
[19]
Sz´ekely, G. J., Rizzo, M. L., Bakirov, N. K., 2007. Measuring and testing dependence by correlation of distances. Ann. Statist. 35, 2769-2794
work page 2007
-
[20]
Sz´ekely, G. J., Rizzo, M. L., 2009. Brownian distance covariance. Ann. Appl. Stat. 3, 1236-1265. 30
work page 2009
-
[21]
Sz´ekely, G. J., Rizzo, M. L., 2013. The distance correlationt-test of independence in high dimension. J. Multivariate Anal. 117, 193-213
work page 2013
-
[22]
Inequalities for therth absolute moment of a sum of random variables, 1≤r≤2
von Bahr, B., Esseen, C., 1965. Inequalities for therth absolute moment of a sum of random variables, 1≤r≤2. Ann. Math. Stat. 36, 299-303
work page 1965
-
[23]
Conditional distance correlation
Wang, X., Pan, W., Hu, W., Tian., Y., Zhang, H., 2015. Conditional distance correlation. J. Amer. Statist. Assoc. 110, 1726-1734
work page 2015
-
[24]
Testing mutual independence in high dimension via distance covariance
Yao, S., Zhang, X., Shao, X., 2018. Testing mutual independence in high dimension via distance covariance. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80, 455-480
work page 2018
-
[25]
Distance-based and RKHS- based dependence metrics in high dimension
Zhu, C., Yao, S., Zhang, X., Shao, X., 2020. Distance-based and RKHS- based dependence metrics in high dimension. Ann. Statist. 48(6), 3366- 3394. 31
work page 2020
discussion (0)
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