Limit theorems for iteration stable tessellations

TL;DR

This paper investigates the large-scale geometric properties of iteration stable tessellations in multi-dimensional space, establishing limit theorems that reveal their asymptotic behavior and distributional limits.

Contribution

It introduces new limit theorems for STIT tessellations, including Gaussian and nonnormal central limit results across different dimensions.

Findings

Gaussian limit theorem for surface increment process

Central limit theorem for total edge length/facet surface

Nonnormal limit distribution in higher dimensions

Abstract

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^d$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.