Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-25T13:02:58.751Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit laws of large numbers for random nearest-neighbour-type graphs

Part of: Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  01 July 2016

Andrew R. Wade
Andrew R. Wade*
Affiliation:
University of Bristol
*
∗ Postal address: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: andrew.wade@bristol.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability13 (2003), 277-303) we give laws of large numbers (in the Lp sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝd, d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Keywords

Nearest-neighbour-type graphlaw of large numbersspanning forestspatial network evolution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F25: $L^p$-limit theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 326 - 342
Copyright
Copyright © Applied Probability Trust 2007 

References

Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau Standards Appl. Math. Ser. 55). U.S. Government Printing Office, Washington, DC.Google Scholar
Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Prob. 3, 1033–1046.Google Scholar
Barndorff-Nielsen, O. and Sobel, M. (1966). On the distribution of the number of admissible points in a vector random sample. Theory Prob. Appl. 11, 249–269.Google Scholar
Berger, N. et al. (2003). Degree distribution of the FKP model. In Automata, Languages and Programming (Lecture Notes Comput. Sci. 2719), eds Baeten, J. C. M., Lenstra, J. K., Parrow, J. and Woeginger, G. J., Springer, Berlin, pp. 725–738.Google Scholar
Bhatt, A. G. and Roy, R. (2004). On a random directed spanning tree. Adv. Appl. Prob. 36, 19–42.Google Scholar
Bickel, P. J. and Breiman, L. (1983). Sums of functions of nearest neighbour distances, moment bounds, and a goodness of fit test. Ann. Prob. 11, 185–214.Google Scholar
Brito, M. R., Quirox, A. J. and Yukich, J. E. (2002). Graph-theoretic procedures for dimension identification. J. Multivariate Anal. 81, 67–84.Google Scholar
Evans, D. and Jones, A. J. (2002). A proof of the Gamma test. Proc. R. Soc. London A 458, 2759–2799.Google Scholar
Evans, D., Jones, A. J. and Schmidt, W. M. (2002). Asymptotic moments of near-neighbour distance distributions. Proc. R. Soc. London A 458, 2839–2849.Google Scholar
Friedman, J. H. and Rafsky, L. C. (1983). Graph-theoretic measures of multivariate association and prediction. Ann. Statist. 2, 377–391.Google Scholar
Gabriel, K. R. and Sokal, R. R. (1969). A new statistical approach to geographic variation analysis. Systematic Zoology 18, 259–278.CrossRefGoogle Scholar
Henze, N. (1987). On the fraction of random points with specified nearest-neighbour interrelations and degree of attraction. Adv. Appl. Prob. 19, 873–895.Google Scholar
Henze, N. and Voigt, B. (1992). Almost sure convergence of certain slowly changing symmetric one- and multi-sample statistics. Ann. Prob. 20, 1086–1098.CrossRefGoogle Scholar
Huang, K. (1987). Statistical Mechanics, 2nd edn. John Wiley, New York.Google Scholar
Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Prob. 6, 495–527.Google Scholar
Kolars, J. F. and Nystuen, J. D. (1974). Human Geography: Spatial Design in World Society. McGraw-Hill, New York.Google Scholar
McGivney, K. (1997). Probabilistic limit theorems for combinatorial optimization problems. , Lehigh University.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.CrossRefGoogle Scholar
Penrose, M. (2003). Random Geometric Graphs (Oxford Studies Prob. 6). Oxford University Press.Google Scholar
Penrose, M. D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33, 1945–1945.Google Scholar
Penrose, M. D. and Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. Appl. Prob. 36, 691–714.Google Scholar
Penrose, M. D. and Wade, A. R. (2006). On the total length of the random minimal directed spanning tree. Adv. Appl. Prob. 38, 336–372.Google Scholar
Penrose, M. D. and Wade, A. R. (2007). Limit theory for the random on-line nearest-neighbour graph. To appear in Random Structures Algorithms.Google Scholar
Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.Google Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.Google Scholar
Percus, A. G. and Martin, O. C. (1998). Scaling universalities of kth nearest neighbor distances on closed manifolds. Adv. Appl. Math. 21, 424–436.Google Scholar
Pielou, E. C. (1977). Mathematical Ecology, 2nd edn. John Wiley, New York.Google Scholar
Smith, W. D. (1989). Studies in computational geometry motivated by mesh generation. , Princeton University.Google Scholar
Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Toussaint, G. (2005). Geometric proximity graphs for improving nearest neighbour methods in instance-based learning and data mining. Internat. J. Comput. Geometry Appl. 15, 101–150.Google Scholar
Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems (Springer Lecture Notes Math. 1675). Springer, Berlin.Google Scholar