Limit theorems for functionals on the facets of stationary random tessellations

TL;DR

This paper establishes limit theorems for the total volumes of manifold processes on the facets of stationary random tessellations, showing asymptotic normality and deriving explicit formulas for mean and variance.

Contribution

It provides new limit theorems for functionals on tessellation facets, including explicit formulas and numerical results for specific tessellation models.

Findings

Total volumes are approximately normally distributed for large windows.

Explicit formulas for mean and variance are derived.

Numerical values are provided for planar Poisson-Voronoi and line tessellations.

Abstract

We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in $W$ are approximately normally distributed when $W$ is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).