Second-Order Properties and Central Limit Theory for the Vertex Process of Iteration Infinitely Divisible and Iteration Stable Random Tessellations in the Plane

TL;DR

This paper analyzes the second-order properties of vertex processes in iteration stable random tessellations in the plane, deriving explicit formulas and a central limit theorem for related geometric functionals.

Contribution

It provides explicit covariance and correlation formulas, variance asymptotics, and a functional central limit theorem for vertex and edge processes in iteration stable tessellations.

Findings

Explicit covariance and pair-correlation functions derived.

Variance formulas for vertex counts in convex windows obtained.

A functional central limit theorem for edge counts and lengths established.

Abstract

The point process of vertices of an iteration infinitely divisible or more specifically of an iteration stable random tessellation in the Euclidean plane is considered. We explicitly determine its covariance measure and its pair-correlation function as well as the cross-covariance measure and the cross-pair-correlation function of the vertex point process and the random length measure in general non-stationary regime, and we specialize it to the stationary and isotropic setting. Exact formulas are given for vertex count variances in compact and convex sampling windows and asymptotic relations are derived. Our results are then compared with those for a Poisson line tessellation having the same length density parameter. Moreover, a functional central limit theorem for the joint process of suitably rescaled total edge count and edge length is established with the process $(\xi,t\xi)$, $t>0,$ arising in the limit, where $\xi$ is a centered Gaussian variable with explicitly known variance.