The Annals of Probability

Limit theorems for iteration stable tessellations

Tomasz Schreiber and Christoph Thäle

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Abstract

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^{d}$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

Article information

Source
Ann. Probab. Volume 41, Number 3B (2013), 2261-2278.

Dates
First available in Project Euclid: 15 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1368623525

Digital Object Identifier
doi:10.1214/11-AOP718

Mathematical Reviews number (MathSciNet)
MR3098072

Zentralblatt MATH identifier
1279.60025

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60J75: Jump processes

Keywords
Central limit theorem functional limit theorem iteration/nesting Markov process martingale theory random tessellation stochastic stability stochastic geometry

Citation

Schreiber, Tomasz; Thäle, Christoph. Limit theorems for iteration stable tessellations. Ann. Probab. 41 (2013), no. 3B, 2261--2278. doi:10.1214/11-AOP718. https://projecteuclid.org/euclid.aop/1368623525.


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