Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws
Abstract
Bergsma--Dassios sign covariance τ* is a rank-based population measure of dependence. Building on zero-characterisation results under specific regularity regimes, we prove that τ*(X,Y)=0 characterises independence for every real-valued bivariate distribution, including mixed and singular laws. For the unnormalised four-sample convention for τ* defined in Subsection 4.3 and the unscaled Blum--Kiefer--Rosenblatt functional ℬ, the proof gives the quantitative inequality τ*≥ 2ℬ. This is a population identification result; no new sample-level limit theorem is claimed. The argument first encodes finite ordered distributions with rational cell probabilities by labelled path trees and applies a nonnegative sum-of-squares representation for a quartet covariance. Rational approximation and nested quantisation then remove all support and regularity restrictions. On finite uniformly weighted label sets, the tree framework also relates an edge-weighted quartet quantity to empirical distance covariance squared. As a separate combinatorial consequence, it yields the asymptotic 2/3 upper bound for the quartet distance between binary phylogenetic trees.
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BibTeX
@misc{grunewald2026bergsmadassios,
title = {{Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws}},
author = {Stefan Grünewald and Libo Huang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30331},
eprint = {2608.30331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30331},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws
AU - Stefan Grünewald
AU - Libo Huang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30331
UR - https://arxiv.org/abs/2608.30331
AB - Bergsma--Dassios sign covariance τ^* is a rank-based population measure of dependence. Building on zero-characterisation results under specific regularity regimes, we prove that τ^*(X,Y)=0 characterises independence for every real-valued bivariate distribution, including mixed and singular laws. For the unnormalised four-sample convention for τ^* defined in Subsection 4.3 and the unscaled Blum--Kiefer--Rosenblatt functional ℬ, the proof gives the quantitative inequality τ^*≥ 2ℬ. This is a population identification result; no new sample-level limit theorem is claimed. The argument first encodes finite ordered distributions with rational cell probabilities by labelled path trees and applies a nonnegative sum-of-squares representation for a quartet covariance. Rational approximation and nested quantisation then remove all support and regularity restrictions. On finite uniformly weighted label sets, the tree framework also relates an edge-weighted quartet quantity to empirical distance covariance squared. As a separate combinatorial consequence, it yields the asymptotic 2/3 upper bound for the quartet distance between binary phylogenetic trees.
ER -