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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.08505 |
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| _version_ | 1866918436997693440 |
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| author | Dietrich, Nicolas Pascal Sánchez, Juan Fernández Trutschnig, Wolfgang |
| author_facet | Dietrich, Nicolas Pascal Sánchez, Juan Fernández Trutschnig, Wolfgang |
| contents | The family $\mathcal{P}_{d}^{λ_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $λ_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements: Working with Iterated Function Systems with Probabi\-lities (IFSPs) we construct measures $μ\in \mathcal{P}_{d}^{λ_{d-1}}$ of the following two types: (i) $μ$ has self-similar fractal support; (ii) $μ$ has self-similar support and models the situation of complete/functional dependence in each direction.As our main results concerning type (i) we prove, firstly, that for every $d\geq 3$ the set $\mathcal{D}_d$ of Hausdorff dimensions of the supports of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ is dense in $[d-1,d]$; and, secondly, that the subset of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ having fractal support is dense in $\mathcal{P}_{d}^{λ_{d-1}}$ with respect to the Wasserstein metric. Moreover, we show the existence of an element in $\mathcal{P}_{3}^{λ_{2}}$ of type (ii) whose support is a Sierpinski tetrahedron and study some generalizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08505 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On d-stochastic measures with fractal support and uniform (d-1)-marginals, and related results Dietrich, Nicolas Pascal Sánchez, Juan Fernández Trutschnig, Wolfgang Probability 28A80 The family $\mathcal{P}_{d}^{λ_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $λ_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements: Working with Iterated Function Systems with Probabi\-lities (IFSPs) we construct measures $μ\in \mathcal{P}_{d}^{λ_{d-1}}$ of the following two types: (i) $μ$ has self-similar fractal support; (ii) $μ$ has self-similar support and models the situation of complete/functional dependence in each direction.As our main results concerning type (i) we prove, firstly, that for every $d\geq 3$ the set $\mathcal{D}_d$ of Hausdorff dimensions of the supports of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ is dense in $[d-1,d]$; and, secondly, that the subset of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ having fractal support is dense in $\mathcal{P}_{d}^{λ_{d-1}}$ with respect to the Wasserstein metric. Moreover, we show the existence of an element in $\mathcal{P}_{3}^{λ_{2}}$ of type (ii) whose support is a Sierpinski tetrahedron and study some generalizations. |
| title | On d-stochastic measures with fractal support and uniform (d-1)-marginals, and related results |
| topic | Probability 28A80 |
| url | https://arxiv.org/abs/2604.08505 |