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July 2005 Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius
Pierre Calka, Tomasz Schreiber
Ann. Probab. 33(4): 1625-1642 (July 2005). DOI: 10.1214/009117905000000134


In this paper, we are interested in the behavior of the typical Poisson–Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson–Voronoi tessellations, convex hulls of Poisson samples and germ–grain models in the unit ball. Besides, we derive analogous facts for the Crofton cell of a stationary Poisson line process in the plane.


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Pierre Calka. Tomasz Schreiber. "Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius." Ann. Probab. 33 (4) 1625 - 1642, July 2005.


Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1084.60008
MathSciNet: MR2150201
Digital Object Identifier: 10.1214/009117905000000134

Primary: 60D05 , 60F10
Secondary: 60G55

Keywords: extreme point , Germ–grain models , large and moderate deviations , Palm distribution , Poisson–Voronoi tessellation , random convex hulls , Stochastic geometry , typical cell

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • July 2005
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