- The paper uses iterated function systems with probabilities to construct singular d-stochastic measures whose every (d−1)-dimensional marginal is uniform.
- It proves that for every d ≥ 3, Hausdorff dimensions of supports are dense in [d−1,d], while conjecturing that every value in this interval occurs.
- It shows fractal measures with non-integer support dimensions are Wasserstein-dense, including completely dependent examples such as a Sierpiński tetrahedron with uniform bivariate marginals.
Background and motivation
A probability measure μ on [0,1]d is d-stochastic if all univariate marginals are uniform; such measures correspond bijectively to d-dimensional copulas. The paper studies the strictly smaller family Pdλd−1 of d-stochastic measures whose every (d−1)-dimensional marginal equals λd−1, the Lebesgue measure on [0,1]d−1. Since copulas are Lipschitz continuous, one might expect their underlying measures to distribute mass regularly; Fredricks, Nelsen and Rodríguez-Lallena already refuted this by constructing doubly stochastic measures whose support has any prescribed Hausdorff dimension s∈(1,2) [FNRL]. The present work extends that result to uniform [0,1]d0-marginals — a substantially stronger constraint — using Iterated Function Systems with Probabilities (IFSPs) built from generalized transformation matrices in the spirit of Trutschnig–Fernández Sánchez [TFS].
IFSP construction and preservation of marginals
The authors fix [0,1]d1 and consider probability distributions [0,1]d2 on [0,1]d3 (generalized transformation matrices) satisfying positive column sums in every coordinate. Each such [0,1]d4 induces a partition of [0,1]d5 into rectangles [0,1]d6, affine contractions [0,1]d7, and an IFSP fulfilling the open set condition. Its Markov operator [0,1]d8 maps [0,1]d9 into itself, so Banach's fixed point theorem yields a unique fixed point d0.
The central technical device is the uniformity condition: for coordinate d1,
d2
Lemma 4.2 shows that if d3 and d4 satisfies this condition w.r.t. coordinate d5, then d6; iterating and passing to the weak limit gives the first main structural theorem: if d7 fulfills the uniformity condition for every coordinate, then all d8-dimensional marginals of d9 equal d0. A corollary adds singularity: whenever some cell has d1, the support of d2 is Lebesgue-null (volume shrinks geometrically under the Hutchinson operator), so d3 is singular w.r.t. d4 while retaining uniform d5-marginals.
Completely dependent measures on self-similar supports
The paper next constructs elements of d6 exhibiting complete dependence in every direction — each coordinate is almost surely a function of the remaining d7 coordinates. A simple non-fractal example is d8 with d9; the authors note they could not locate this example in the literature despite suspecting it is known.
The main construction uses matrices Pdλd−10 (equal side lengths Pdλd−11, uniformity in all coordinates), which induce IFSPs consisting entirely of similarities with factor Pdλd−12. For
Pdλd−13
the resulting measure Pdλd−14 satisfies three properties simultaneously: it lies in Pdλd−15; it is singular with self-similar support of Hausdorff dimension exactly Pdλd−16 (Pdλd−17 similarities of factor Pdλd−18); and it is completely dependent in each direction. Complete dependence is established via the metric Pdλd−19 based on Markov kernels: the family of completely dependent measures is closed in d0, d1 is a contraction there, and each iterate of a completely dependent seed remains completely dependent because for every d2 exactly one d3 has d4.
The case d5, d6 is particularly striking: the support of d7 is a Sierpinski tetrahedron, yet the measure has uniform bivariate marginals and is completely dependent in each direction. This demonstrates that even maximal dependence structure is compatible with highly fractal support under strong marginal constraints. A second family of examples, built from cyclic permutations of d8, extends the construction to all d9.
Density of fractal dimensions
Combining convexity of (d−1)0 with permutation-invariance arguments, the paper proves its first main result:
Theorem. For every (d−1)1, the set (d−1)2 is dense in (d−1)3.
The proof averages permutations (d−1)4 of the base matrix (d−1)5: the convex combination (d−1)6 induces an IFSP whose number of similarities grows from (d−1)7 to at most (d−1)8 in steps bounded by (d−1)9, giving Hausdorff dimensions λd−10 that hit every interval λd−11. Since λd−12 is dense in λd−13, density follows. An explicit three-dimensional example achieves dimension λd−14 with uniform bivariate marginals. The authors conjecture the stronger statement λd−15 but leave it open.
Topological prevalence of pathological copulas
The second main result strengthens the picture from existence to genericity:
Theorem. For every λd−16, the family λd−17 of elements of λd−18 whose support has non-integer Hausdorff dimension is dense in λd−19, where [0,1]d−10 is the Wasserstein-1 (Hutchinson) metric.
In fact, a remark records the stronger statement that for every open interval [0,1]d−11, the elements whose support dimension lies in [0,1]d−12 are dense. The proof runs the checkerboard approximations of Mikusiński–Taylor in reverse: given [0,1]d−13, the discretized matrix [0,1]d−14 belongs to [0,1]d−15, and [0,1]d−16 converges to [0,1]d−17 uniformly on copulas. Replacing the seed [0,1]d−18 by any [0,1]d−19 with fractal support preserves both membership in s∈(1,2)0 (by the main structural theorem) and the Hausdorff dimension of the support (bi-Lipschitz invariance plus countable stability), while Lipschitz continuity of copulas controls the discretization error within s∈(1,2)1 per term. Consequently, arbitrarily close to any element of s∈(1,2)2 — including the product measure s∈(1,2)3 itself — sit singular measures with fractal, even completely-dependent-type, structure.
Limitations and open questions
Several caveats qualify these results. First, the density claim for s∈(1,2)4 falls short of surjectivity onto s∈(1,2)5; whether every value in the interval is attained is explicitly conjectured but unresolved. Second, the constructions require the uniformity condition and, for the similarity-based results, the symmetric setting s∈(1,2)6; nothing is claimed about supports of non-self-similar type or about the extreme points of s∈(1,2)7, whose characterization (even for s∈(1,2)8) remains open. Third, the mod-s∈(1,2)9 sum example of complete dependence is presented without a literature reference, and the authors concede they could not verify whether it has appeared before. Finally, all results concern [0,1]d00; the two-dimensional analogue with prescribed one-dimensional marginals is just [0,1]d01 itself, where the FNR result applies but the [0,1]d02-marginal machinery does not.
Conclusion
The paper establishes that the constrained family [0,1]d03 is as pathological as unconstrained [0,1]d04: it contains singular measures with self-similar supports of dense-set-many Hausdorff dimensions in [0,1]d05, contains completely dependent elements supported on fractals — including a Sierpinski-tetrahedron-supported [0,1]d06-stochastic measure with uniform bivariate marginals — and its fractal members are Wasserstein-dense. These findings sharpen the FNR phenomenon under much stronger marginal constraints and provide an effective IFSP toolkit for constructing explicit examples.