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 $d\ge 3$ 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.

18 pages