Open Access
May 2013 Limit theorems for iteration stable tessellations
Tomasz Schreiber, Christoph Thäle
Ann. Probab. 41(3B): 2261-2278 (May 2013). DOI: 10.1214/11-AOP718


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.


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Tomasz Schreiber. Christoph Thäle. "Limit theorems for iteration stable tessellations." Ann. Probab. 41 (3B) 2261 - 2278, May 2013.


Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1279.60025
MathSciNet: MR3098072
Digital Object Identifier: 10.1214/11-AOP718

Primary: 60D05 , 60F17
Secondary: 60F05 , 60J75

Keywords: central limit theorem , Functional limit theorem , iteration/nesting , Markov process , Martingale theory , random tessellation , Stochastic geometry , Stochastic stability

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 3B • May 2013
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