STIT Tessellations -- Ergodic Limit Theorems and Bounds for the Speed of Convergence

TL;DR

This paper investigates the statistical properties of homogeneous STIT tessellations in Euclidean space, deriving bounds on variance and applying ergodic theorems to understand their long-term behavior.

Contribution

It provides new variance bounds for additive functionals and applies ergodic theorems to subadditive functionals in the context of STIT tessellations.

Findings

Derived an upper bound for the variance of additive functionals.

Applied ergodic theorems to subadditive functionals.

Extended understanding of the statistical properties of STIT tessellations.

Abstract

We consider homogeneous STIT tessellations in the $\ell$-dimensional Euclidean space ${\mathbb R}^\ell$. Based on results for the spatial $\beta$-mixing coefficient an upper bound for the variance of additive functionals of tessellations is derived, using results by Yoshihara and Heinrich. Moreover, ergodic theorems are applied to subadditive functionals.