The Annals of Probability

Limit theorems for iteration stable tessellations

Tomasz Schreiber and Christoph Thäle

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

First available in Project Euclid: 15 May 2013

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Zentralblatt MATH identifier

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

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


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

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