发表机构
Shanghai Institute of Nutrition and Health, Chinese Academy of Science（中国科学院上海营养与健康研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明Bergsma--Dassios符号协方差τ*=0可刻画任意实值二元分布的独立性，通过标记路径树等方法推导定量不等式，还得到二元系统发育树四分距离的渐近2/3上界。
AI 中文摘要
Bergsma--Dassios符号协方差τ*是一种基于秩的总体相关性度量。基于特定正则性条件下的零刻画结果，我们证明τ*(X,Y)=0可刻画所有实值二元分布的独立性，包括混合分布和奇异分布。对于第4.3节定义的τ*的未归一化四样本约定以及未缩放的Blum--Kiefer--Rosenblatt泛函B，该证明给出定量不等式τ*≥2B。这是一个总体识别结果，未提出新的样本水平极限定理。论证首先用带有理单元概率的标记路径树编码有限有序分布，并应用四分协方差的非负平方和表示；再通过有理逼近和嵌套量化消除所有支撑和正则性限制。在有限均匀加权标签集上，树框架还将边加权四分量与经验距离协方差平方关联；作为独立的组合结论，它给出二元系统发育树间四分距离的渐近2/3上界。
英文摘要
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 $\mathscr {B}$, the proof gives the quantitative inequality $τ^*\ge 2\mathscr {B}$. 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.
