Length distributions of edges in planar stationary and isotropic STIT tessellations. (English. Russian original) Zbl 1155.60005

Mecke, J.; Nagel, W.; Weiss, V.

J. Contemp. Math. Anal., Armen. Acad. Sci. 42, No. 1, 28-43 (2007); translation from Izv. Nats. Akad. Nauk Armen., Mat. 42, No. 1, 39-60 (2007).

The iteration operation for a planar tessellation is defined by subdividing its cells using a sequence of further iid tessellations. If the rescaled result of this (possibly iterated) operation coincides in distribution with the original tessellation \(\mathcal{Y}\), then \(\mathcal{Y}\) is said to be stable with respect to iteration (STIT) [see W. Nagel and V. Weiss, Adv. Appl. Probab. 37, 859–883 (2005; Zbl 1098.60012)].
The authors recall the construction of STIT tessellations in bounded windows, discuss properties of mean values and then proceed to study length distributions for segments of three different types that appear as edges of STIT tessellations. The authors derive explicit formulae for their length distributions in the stationary and isotropic case and discuss various properties of the obtained length distributions.

Reviewer: Ilya S. Molchanov (Bern)

MSC:

60D05	Geometric probability and stochastic geometry
52A22	Random convex sets and integral geometry (aspects of convex geometry)

Keywords:

random tessellation; iteration of tessellations; stable tessellation; edge

Citations:

Zbl 1098.60012

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