Advances in Applied Probability

Extremes for the inradius in the Poisson line tessellation

Nicolas Chenavier and Ross Hemsley

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A Poisson line tessellation is observed in the window Wρ := B(0, π-1/2ρ1/2) for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using the Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to ∞. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 2 (2016), 544-573.

Dates
First available in Project Euclid: 9 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1465490762

Mathematical Reviews number (MathSciNet)
MR3511775

Zentralblatt MATH identifier
1342.60011

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 62G32: Statistics of extreme values; tail inference

Keywords
Line tessellation Poisson point process extreme value order statistic

Citation

Chenavier, Nicolas; Hemsley, Ross. Extremes for the inradius in the Poisson line tessellation. Adv. in Appl. Probab. 48 (2016), no. 2, 544--573. https://projecteuclid.org/euclid.aap/1465490762


Export citation

References

  • Beermann, M., Redenbach, C. and Thäle, C. (2014). Asymptotic shape of small cells. Math. Nachr. 287, 737–747.
  • 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. Appl. Prob. 35, 551–562.
  • Calka, P. and Chenavier, N. (2014). Extreme values for characteristic radii of a Poisson–Voronoi tessellation. Extremes 17, 359–385.
  • Charikar, M. S. (2002). Similarity estimation techniques from rounding algorithms. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 380–388.
  • Chenavier, N. (2014). A general study of extremes of stationary tessellations with examples. Stoch. Process. Appl. 124, 2917–2953.
  • De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • Goudsmit, S. (1945). Random distribution of lines in a plane. Rev. Modern Phys. 17, 321–322.
  • Heinrich, L. (2009). Central limit theorems for motion-invariant Poisson hyperplanes in expanding convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. 81, 187–212.
  • Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Prob. 16, 919–950.
  • Hsing, T. (1988). On the extreme order statistics for a stationary sequence. Stoch. Process. Appl. 29, 155–169.
  • Hug, D. and Schneider, R. (2014). Approximation properties of random polytopes associated with Poisson hyperplane processes. Adv. Appl. Prob. 46, 919–936.
  • Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32, 1140–1167.
  • Ju, L., Gunzburger, M. and Zhao, W. (2006). Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi–Delaunay triangulations. SIAM J. Sci. Comput. 28, 2023–2053.
  • Leadbetter, M. R. (1974). On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.
  • Leadbetter, M. R. and Rootzén, H. (1998). On extreme values in stationary random fields. In Stochastic Processes and Related Topics, Birkhäuser, Boston, MA, pp. 275–285.
  • Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 901–907.
  • Miles, R. E. (1964). Random polygons determined by random lines in a plane. II. Proc. Nat. Acad. Sci. USA 52, 1157–1160.
  • Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes (Monogr. Statist. Appl. Prob. 100). Chapman & Hall/CRC, Boca Raton, FL.
  • Penrose, M. (2003). Random Geometric Graphs. Oxford Univeristy Press.
  • Plan, Y. and Vershynin, R. (2014). Dimension reduction by random hyperplane tessellations. Discrete Comput. Geometry 51, 438–461.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Santaló, L. A. (2004). Integral Geometry and Geometric Probability, 2nd edn. Cambridge University Press.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Schulte, M. and Thäle, C. (2012). The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stoch. Process. Appl. 122, 4096–4120.
  • Schulte, M. and Thäle, C. (2016). Poisson point process convergence and extreme values in stochastic geometry. Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry, Springer, Cham.