Fractal random series generated by Poisson-Voronoi tessellations

TL;DR

This paper introduces a new class of fractal functions generated by Poisson-Voronoi tessellations, analyzing their fractal dimensions and distributional properties, extending classical models with random partitions.

Contribution

It constructs a novel family of fractal random series based on Poisson-Voronoi tessellations, providing explicit dimension calculations and new distributional insights.

Findings

The fractal dimension of the graph is explicitly calculated.

Distributional properties of Poisson-Voronoi tessellations are characterized.

A deterministic Takagi-Knopp series with hexagonal bases is analyzed.

Abstract

In this paper, we construct a new family of random series defined on $\R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\R^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tessellation on the one hand, an estimate of the oscillations of the function coupled with an application of a Frostman-type lemma on the other hand. Finally, we introduce two related models and provide in particular a box-dimension calculation for a derived deterministic Takagi-Knopp series with hexagonal bases.