The Annals of Probability

Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius

Pierre Calka and Tomasz Schreiber

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Abstract

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.

Article information

Source
Ann. Probab., Volume 33, Number 4 (2005), 1625-1642.

Dates
First available in Project Euclid: 1 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1120224593

Digital Object Identifier
doi:10.1214/009117905000000134

Mathematical Reviews number (MathSciNet)
MR2150201

Zentralblatt MATH identifier
1084.60008

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F10: Large deviations
Secondary: 60G55: Point processes

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

Citation

Calka, Pierre; Schreiber, Tomasz. Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius. Ann. Probab. 33 (2005), no. 4, 1625--1642. doi:10.1214/009117905000000134. https://projecteuclid.org/euclid.aop/1120224593


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