Annals of Probability

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

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

https://projecteuclid.org/euclid.aop/1120224593

Digital Object Identifier
doi:10.1214/009117905000000134

Mathematical Reviews number (MathSciNet)
MR2150201

Zentralblatt MATH identifier
1084.60008

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

References

• Baccelli, F. and Błaszczyszyn, B. (2001). On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications. Adv. in Appl. Probab. 33 293–323.
• Baccelli, F., Klein, M., Lebourges, M. and Zuyev, Z. (1997). Stochastic geometry and architecture of communication networks. J. Telecommunication Systems 2 209–227.
• Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random convex hull in a bounded convex set. Probab. Theory Related Fields 111 517–550.
• Calka, P. (2002). The distributions of the smallest disks containing the Poisson–Voronoi typical cell and the Crofton cell in the plane. Adv. in Appl. Probab. 34 702–717.
• Calka, P. (2003). An explicit expression of the distribution of the number of sides of the typical Poisson–Voronoi cell. Adv. in Appl. Probab. 35 863–870.
• Calka, P. (2003). Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson–Voronoi tessellation and a Poisson line process. Adv. in Appl. Probab. 35 551–562.
• Gerstein, M., Tsai, J. and Levitt, M. (1995). The volume of atoms on the protein surface: Calculated from simulation, using Voronoi polyhedra. Journal of Molecular Biology 249 955–966.
• Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33 958–972.
• Goldman, A. (1998). Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Probab. 26 1727–1750.
• Groeneboom, P. (1988). Limit theorems for convex hulls. Probab. Theory Related Fields 79 327–368.
• Hug, D., Reitzner, M. and Schneider, R. (2004). Large Poisson–Voronoi cells and Crofton cells. Adv. in Appl. Probab. 36 667–690.
• Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane process. Ann. Probab. 32 1140–1167.
• Kovalenko, I. N. (1999). A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons. J. Appl. Math. Stochastic Anal. 12 301–310.
• Küfer, K.-H. (1994). On the approximation of a ball by random polytopes. Adv. in Appl. Probab. 26 876–892.
• Kumar, S. and Singh, R. N. (1995). Thermal conductivity of polycrystalline materials. J. Amer. Ceramics Soc. 78 728–736.
• Massé, B. (2000). On the LLN for the number of vertices of a random convex hull. Adv. in Appl. Probab. 32 675–681.
• Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8.
• Melkemi, M. and Vandorpe, D. (1994). Voronoi diagrams and applications. In Eighth Canadian Conference on Image Processing and Pattern Recognition 88–95. CIPPR Society, Banff, Alberta.
• Miles, R. E. (1973). The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10 256–290.
• Møller, J. (1994). Lectures on Random Voronoi Tessellations. Springer, New York.
• Nagaev, A. V. (1995). Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain. Ann. Inst. Statist. Math. 47 21–29.
• Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. Wiley, Chichester.
• Reitzner, M. (2003). Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31 2136–2166.
• Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von $n$ zufällig gewählten Punkten. Z. Wahrsch. Verw. Gebiete 2 75–84.
• Schneider, R. (1988). Random approximation of convex sets. J. Microscopy 151 211–227.
• Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge Univ. Press.
• Schreiber, T. (2000). Large deviation principle for set-valued union processes. Probab. Math. Statist. 20 273–285.
• Schreiber, T. (2002). Limit theorems for certain functionals of unions of random closed sets. Teor. Veroyatnost. i Primenen. 47 130–142.
• Schreiber, T. (2002). Variance asymptotics and central limit theorems for volumes of unions of random closed sets. Adv. in Appl. Probab. 34 520–539.
• Schreiber, T. (2003). Asymptotic geometry of high density smooth-grained Boolean models in bounded domains. Adv. in Appl. Probab. 35 913–936.
• Schreiber, T. (2003). A note on large deviation probabilities for volumes of unions of random closed sets. Available at http://www.mat.uni.torun.pl/preprints.
• Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. Wiley, Chichester.
• Zuyev, S. A. (1992). Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures Algorithms 3 149–162.