On d-stochastic measures with fractal support and uniform (d-1)-marginals, and related results

Abstract

The family $\mathcal{P}_{d}^{\lambda_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $\lambda_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements: Working with Iterated Function Systems with Probabi\-lities (IFSPs) we construct measures $\mu \in \mathcal{P}_{d}^{\lambda_{d-1}}$ of the following two types: (i) $\mu$ has self-similar fractal support; (ii) $\mu$ 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 $\mathcal{P}_{d}^{\lambda_{d-1}}$ is dense in $[d-1,d]$; and, secondly, that the subset of elements in $\mathcal{P}_{d}^{\lambda_{d-1}}$ having fractal support is dense in $\mathcal{P}_{d}^{\lambda_{d-1}}$ with respect to the Wasserstein metric. Moreover, we show the existence of an element in $\mathcal{P}_{3}^{\lambda_{2}}$ of type (ii) whose support is a Sierpinski tetrahedron and study some generalizations.