The Annals of Applied Probability

Central Limit Theorems for Some Graphs in Computational Geometry

Mathew D. Penrose and J.E. Yukich

Full-text: Open access


Let $(B_n)$ be an increasing sequence of regions in $d$ -dimensional space with volume $n$ and with union $\mathbb{R}^d$. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to $(B_n)$, or by by taking $n$ uniformly distributed points in $(B_n)$. The sets $(B_n)$ could be all cubes but a more general class of regions$(B_n)$ is considered. Using this general result we obtain central limit theorems for specific functionals suchas total edge lengthand number of components, defined in terms of graphs such as the $k$-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.

Article information

Ann. Appl. Probab., Volume 11, Number 4 (2001), 1005-1041.

First available in Project Euclid: 5 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 60F05
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Central limit theorems computational geometry $k$-nearest neighbors graph sphere of influence graph Voronoi graph.


Penrose, Mathew D.; Yukich, J.E. Central Limit Theorems for Some Graphs in Computational Geometry. Ann. Appl. Probab. 11 (2001), no. 4, 1005--1041. doi:10.1214/aoap/1015345393.

Export citation


  • [1] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033-1046.
  • [2] Bickel, P. J. and Breiman, L. (1983). Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 185-214.
  • [3] Devroye, L. (1988). The expected size of some graphs in computational geometry. Comput. Math. Appl. 15 53-64.
  • [4] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth& Brooks/Cole, Pacific Grove, CA.
  • [5] F ¨uredi, Z. (1997). The expected size of the sphere of influence graph. Bolyai Soc. Math. Stud. 6 ("Intuitive Geometry Budapest 1995") 319-326.
  • [6] Groeneboom, P. (1988). Limit theorems for convex hulls. Probab. Theory Related Fields 79 327-368.
  • [7] H¨aggstr ¨om, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 295-315.
  • [8] Heinrich, L. (1994). Normal approximation for some mean value estimates of absolutely regular tessellations. Math. Methods Statist. 3 1-24.
  • [9] Henze, N. (1987). On the fraction of random points with specified nearest-neighbour interactions and degree of attraction. Adv. in Appl. Probab. 19 873-895.
  • [10] Hitczenko, P., Janson, S. and Yukich, J. E. (1999). On the variance of the random sphere of influence graph. Random Structures Algorithms 14 139-152.
  • [11] Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
  • [12] Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees I. Ann. Appl. Probab. 7 996-1020.
  • [13] Lee, S. (1999). The central limit theorem for Euclidean minimal spanning trees II. Adv. in Appl. Probab. 31 969-984.
  • [14] McGivney, K. and Yukich, J. E. (1999). Asymptotics for Voronoi tessellations on random samples. Stochastic Process. Appl. 83 273-288.
  • [15] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620-628.
  • [16] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Univ. Press.
  • [17] Michael, T. S. and Quint, T. (1994). Sphere of influence graphs: a survey. Congr. Numer. 105 153-160.
  • [18] Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester.
  • [19] Penrose, M. D. (2001). A spatial central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Probab. 29.
  • [20] Penrose, M. D. and Yukich, J. E. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12.
  • [21] Preparata, F. P. and Shamos, M. I. (1985). Computational Geometry: an Introduction. Springer, New York.
  • [22] Toussaint, G. T. (1982). Computational geometric problems in pattern recognition. In Pattern Recognition Theory and Applications (J. Kittler, K. S. Fu and L. F. Pau, eds.), Reidel, Dordrecht. 73-91.
  • [23] Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.