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 di...
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Veröffentlicht in: | The Annals of probability 2005-07, Vol.33 (4), p.1625-1642 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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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ISSN: | 0091-1798 2168-894X |
DOI: | 10.1214/009117905000000134 |