2005/07/01 by Pierre Calka, Tomasz Schreiber
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Random Matrices and Applications #math.PR #msc:60D05 #msc:60F10 #msc:60G55
paper · pdf · doi:10.1214/009117905000000134
published as Annals of Probability 2005, Vol. 33, No. 4, 1625-1642 · Published at http://dx.doi.org/10.1214/009117905000000134 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/07/01 · arxiv created 2005/07/22 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.