Purely Singular Copulas Are Typical for the Markov-Kernel Metric D₁

Durante, Fernández-Sánchez and Trutschnig (Oberwolfach Report 20/2015) asked whether the class of purely singular \(d\)-dimensional copulas is of second category in \((\mathcal{C}_d,D_1)\), where \(D_1\) is the metric of Trutschnig defined through the conditional distributions (Markov kernels) of the copulas. The answer is yes: for every \(d\ge2\) the purely singular copulas form a dense \(G_\delta\) set in \((\mathcal{C}_d,D_1)\), hence a co-meager set. This follows in a few lines from known results, and we claim no priority for it: the purely singular copulas form a \(G_\delta\) set already for the coarser uniform metric, \((\mathcal{C}_d,D_1)\) is complete, and the check-min approximation \(B_N(A)\) of a copula \(A\), which carries the mass of each cell of the \(N\)-grid uniformly on the main diagonal of the cell, is purely singular and tends to \(A\) in \(D_1\). For \(d=2\) this density is classical; Durante, Fernández-Sánchez and Trutschnig use it in “A typical copula is singular” (2015), citing Mikusiński and Taylor (Metrika 2010), and with Theorem 4.1 of that paper it settles the case \(d=2\), which the paper nevertheless lists as open. For \(d\ge3\) the density may also be in Mikusiński and Taylor (2010), a paper on approximations of \(n\)-copulas that we could not read; we prove it with the bound \(D_1(A,B_N(A))\)\({}\le\lVert H_A\)\({}-P_NH_A\rVert_1\)\({}+(d\)\({}-1)\)\(/N\), where \(H_A\) is the conditional distribution function and \(P_N\) averages over the columns of the grid. New, to our knowledge, is that the copulas whose conditional distributions are, for almost every value of the conditioning coordinate, singular and without atoms also form a dense \(G_\delta\) set in \((\mathcal{C}_d,D_1)\): in this metric a typical copula has diffuse singular conditional distributions, whereas for \(d=2\) and the uniform metric they are typically point masses. If \(D_1\) is replaced by the \(L^1\) distance of the copula functions, as in the restatement of the question in version 1.7.0 of the UnsolvedMath collection, the question reduces to the known result for the uniform metric. This is an unrefereed note.