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

TL;DR

This paper constructs and analyzes pathological probability measures with fractal supports and uniform marginals, revealing dense sets of such measures with complex Hausdorff dimensions and specific fractal structures.

Contribution

It demonstrates the density of measures with fractal supports and specific Hausdorff dimensions within the family of measures with uniform marginals, using iterated function systems.

Findings

Hausdorff dimensions of supports are dense in [d-1, d] for d ≥ 3

Measures with fractal support are dense in the Wasserstein metric

Existence of a measure with a Sierpinski tetrahedron support in three dimensions

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.