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Asymptotic Methods for Random Tessellations

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Stochastic Geometry, Spatial Statistics and Random Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2068))

Abstract

In this chapter, we are interested in two classical examples of random tessellations which are the Poisson hyperplane tessellation and Poisson–Voronoi tessellation. The first section introduces the main definitions, the application of an ergodic theorem and the construction of the so-called typical cell as the natural object for a statistical study of the tessellation. We investigate a few asymptotic properties of the typical cell by estimating the distribution tails of some of its geometric characteristics (inradius, volume, fundamental frequency). In the second section, we focus on the particular situation where the inradius of the typical cell is large. We start with precise distributional properties of the circumscribed radius that we use afterwards to provide quantitative information about the closeness of the cell to a ball. We conclude with limit theorems for the number of hyperfaces when the inradius goes to infinity.

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  1. Université de Rouen, Rouen, France

    Pierre Calka

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  1. Pierre Calka
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Correspondence to Pierre Calka .

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  1. Helmholtzstr. 18, Ulm, 89069, Germany

    Evgeny Spodarev

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Calka, P. (2013). Asymptotic Methods for Random Tessellations. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_6

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