Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds

Abstract

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 $ \tau^{*}(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 $\tau^{*}=12D+24R$ can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions $D_{\epsilon,\eta}$ and $R_{\epsilon,\eta}$, we establish the sharp comparisons $ 9\widetilde D\geq\sum_{\epsilon,\eta}D_{\epsilon,\eta}, \qquad 9\widetilde R\geq\sum_{\epsilon,\eta}R_{\epsilon,\eta}. $ Consequently, $\tau^{*}\geq0$ for every bivariate law, and the Bergsma--Dassios conjecture is settled: $\tau^{*}=0$ if and only if $X\perp\!\!\!\perp Y$. The boundary comparisons also yield the universal bound $\tau^{*}\geq(8/3)R$. 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 $\tau^{*}\geq24\widetilde R$ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.