AI 中文总结
本文证明了 Bergsma--Dassios 符号协方差非负性猜想，通过平局对称化分解为两个非负分量，并给出锐利不等式界。
AI 中文摘要
我们证明了 Bergsma 和 Dassios 的猜想，即他们的符号协方差 $\tau^{*}$ 对每个二元分布都是非负的，并且仅在独立时为零。先前的结果覆盖了离散和绝对连续分布、它们的混合以及具有无原子边缘的分布。我们构造了未归一化的 Hoeffding 和 Blum--Kiefer--Rosenblatt 泛函 $D$ 和 $R$ 的平局对称化扩展 $\widetilde D$ 和 $\widetilde R$。对于无原子边缘，它们等于 $D$ 和 $R$，而对于任意分布，它们是非负的且在独立时为零。将符号协方差核的四个参数分成两对，得到一个配对核，其 Hoeffding 投影唯一地决定了分解为两个非负分量 $ \tau^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qquad \widetilde R(X,Y)=0\quad\Longleftrightarrow\quad X\perp\\!\\!\\!\perp Y. $ 对于四个严格/非严格边界版本 $D_{\epsilon,\eta}$ 和 $R_{\epsilon,\eta}$，其中 $\epsilon,\eta\in\{<,\leq\}$，锐界为 $ 9\widetilde D\geq\sum_{\epsilon,\eta}D_{\epsilon,\eta}, \qquad 9\widetilde R\geq\sum_{\epsilon,\eta}R_{\epsilon,\eta}. $ 一个精确的平方和证书证明了有限表格情形，弱序量化将其转移到任意分布。每个界中的因子 $9$ 和 $\tau^{*}\geq24\widetilde R$ 中的系数 $24$ 是锐利的。
英文摘要
We construct tie-symmetrised extensions $\widetilde D$ and $\widetilde R$ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals $D$ and $R$ and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ τ^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qquad \widetilde D,\widetilde R\geq0, \qquad \widetilde R(X,Y)=0\quad\Longleftrightarrow\quad X\perp\!\!\!\perp Y. $ For atomless margins the components reduce to $D$ and $R$, whereas in the presence of ties the corresponding classical identity $τ^{*}=12D+24R$ can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions $D_{ε,η}$ and $R_{ε,η}$, we establish the sharp comparisons $ 9\widetilde D\geq\sum_{ε,η}D_{ε,η}, \qquad 9\widetilde R\geq\sum_{ε,η}R_{ε,η}. $ Consequently, $τ^{*}\geq0$ for every bivariate law, and the Bergsma--Dassios conjecture is settled: $τ^{*}=0$ if and only if $X\perp\!\!\!\perp Y$. The boundary comparisons also yield the universal bound $τ^{*}\geq(8/3)R$. Finite-table arguments, including an exact sum-of-squares certificate, establish the boundary inequalities, and weak-order quantisation transfers them to arbitrary laws. The factor $9$ in each comparison and the coefficient $24$ in $τ^{*}\geq24\widetilde R$ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.
Comments33 pages, including supplementary proofs; Python/SymPy verification scripts included as ancillary files. Revised title, abstract, introduction, and exposition; expanded discussion of consequences and updated references. Main theorem unchanged
