Intrinsic Volumes of the Maximal Polytope Process in Higher Dimensional STIT Tessellations

TL;DR

This paper investigates the intrinsic volumes of maximal polytopes in higher-dimensional STIT tessellations, deriving formulas for mean, variance, and covariance, and establishing a multivariate limit theorem with non-Gaussian limits.

Contribution

It extends the analysis of STIT tessellations to higher dimensions, providing new formulas and a limit theorem for intrinsic volumes in dimensions three and above.

Findings

Formulas for mean values, variances, and covariance measures of maximal polytopes.

A multivariate limit theorem with non-Gaussian limits for rescaled intrinsic volumes.

Distinct behavior in higher dimensions compared to the planar case.

Abstract

Stationary and isotropic iteration stable random tessellations are considered, which can be constructed by a random process of cell division. The collection of maximal polytopes at a fixed time $t$ within a convex window $W\subset{\Bbb R}^d$ is regarded and formulas for mean values, variances, as well as a characterization of certain covariance measures are proved. The focus is on the case $d\geq 3$, which is different from the planar one, treated separately in \cite{ST2}. Moreover, a multivariate limit theorem for the vector of suitably rescaled intrinsic volumes is established, leading in each component -- in sharp contrast to the situation in the plane -- to a non-Gaussian limit.