## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 42, Number 4 (2010), 913-935.

### Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane

Tomasz Schreiber and Christoph Thäle

#### 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-correlation function of the vertex point process and the random length
measure in the general nonstationary regime. We also give special formulae in
the stationary and isotropic setting. Exact formulae 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
counts and edge lengths is established with the process
(ξ, *t*ξ), *t* > 0, arising in the limit, where
ξ is a centered Gaussian variable with explicitly known variance.

#### Article information

**Source**

Adv. in Appl. Probab. Volume 42, Number 4 (2010), 913-935.

**Dates**

First available in Project Euclid: 23 December 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1293113144

**Digital Object Identifier**

doi:10.1239/aap/1293113144

**Mathematical Reviews number (MathSciNet)**

MR2796670

**Zentralblatt MATH identifier**

1221.60008

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 60G55: Point processes 60J75: Jump processes 60F05: Central limit and other weak theorems

**Keywords**

Central limit theorem covariance measure cross correlation Markov process iteration/nesting martingale pair-correlation function random tessellation stochastic stability stochastic geometry

#### Citation

Schreiber, Tomasz; Thäle, Christoph. Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv. in Appl. Probab. 42 (2010), no. 4, 913--935. doi:10.1239/aap/1293113144. https://projecteuclid.org/euclid.aap/1293113144