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

Abstract

We construct tie-symmetrised extensions $\tilde{D}$ and $\tilde{R}$ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals $D$ and $R$ and prove, for every real-valued bivariate law, the exact nonnegative decomposition ${\tau }^{\ast }(X,Y)=12\tilde{D}(X,Y)+24\tilde{R}(X,Y),\tilde{D},\tilde{R}\ge 0,\tilde{R}(X,Y)=0⟺X\perp \perp Y.$ For atomless margins the components reduce to $D$ and $R$, whereas in the presence of ties the corresponding classical identity ${\tau }^{\ast }=12D+24R$ 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 $9\tilde{D}\ge {\sum }_{\epsilon ,\eta }{D}_{\epsilon ,\eta },9\tilde{R}\ge {\sum }_{\epsilon ,\eta }{R}_{\epsilon ,\eta }.$ Consequently, ${\tau }^{\ast }\ge 0$ for every bivariate law, and the Bergsma--Dassios conjecture is settled: ${\tau }^{\ast }=0$ if and only if $X\perp \perp Y$. The boundary comparisons also yield the universal bound ${\tau }^{\ast }\ge (8/3)R$. 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 ${\tau }^{\ast }\ge 24\tilde{R}$ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.

33 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