The Annals of Probability
- Ann. Probab.
- Volume 33, Number 4 (2005), 1625-1642.
Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius
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.
Ann. Probab., Volume 33, Number 4 (2005), 1625-1642.
First available in Project Euclid: 1 July 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F10: Large deviations
Secondary: 60G55: Point processes
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