Laws of Large Numbers and Nearest Neighbor Distances



We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for large n, correcting some inaccuracies in the literature on the way. Motivation comes partly from the problem of consistent estimation of certain entropies of f.


Minimal Span Tree Tsallis Entropy Multivariate Density Unbounded Support Gaussian Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Research of Matthew Penrose supported in part by the Alexander von Humboldt Foundation through a Friedrich Wilhelm Bessel Research Award. Research of J.E. Yukich supported in part by NSF grant DMS-0805570.


  1. 1.
    Baryshnikov Y, Yukich JE (2005) Gaussian limits for random measures in geometric probability. Ann Appl Probab 15:213–253MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beirlant J, Dudewicz E, Györfi L, Meulen E (1997) Non-parametric entropy estimation: An overview. Int J Math Stat Sci 6(1):17–39MATHMathSciNetGoogle Scholar
  3. 3.
    Bickel P, Breiman L (1983) Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann Probab 11:185–214MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Costa J, Hero III A (2006) Determining intrinsic dimension and entropy of high-dimensional shape spaces. In: Krim H, Yezzi A (eds) Statistics and analysis of shapes. Birkhäuser, Basel, pp 231–252CrossRefGoogle Scholar
  5. 5.
    Evans D, Jones AJ, Schmidt WM (2002) Asymptotic moments of near-neighbour distance distributions. Proc R Soc Lond A Math Phys Sci 458:2839–2849MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feller W (1971) An Introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  7. 7.
    Havrda J, Charvát F (1967) Quantification method of classification processes. Concept of structural α-entropy. Kybernetika (Prague) 3:30–35MATHGoogle Scholar
  8. 8.
    Jiménez R, Yukich JE (2002) Strong laws for Euclidean graphs with general edge weights. Stat Probab Lett 56:251–259MATHCrossRefGoogle Scholar
  9. 9.
    Kozachenko LF, Leonenko NN (1987) A statistical estimate for the entropy of a random vector. Probl Inf Transm 23:95–101MATHMathSciNetGoogle Scholar
  10. 10.
    Leonenko NN, Pronzato L, Savani V (2008) A class of Rényi information estimators for multidimensional densities. Ann Stat 36:2153–2182MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Loftsgaarden DO, Quesenberry CP (1965) A nonparametric estimate of a multivariate density function. Ann Math Stat 36:1049–1051MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Penrose MD (2007a) Gaussian limits for random geometric measures. Electron J Probab 12:989–1035MATHMathSciNetGoogle Scholar
  13. 13.
    Penrose MD (2007b) Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13:1124–1150MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Penrose MD, Yukich JE (2001) Central limit theorems for some graphs in computational geometry. Ann Appl Probab 11:1005–1041MATHMathSciNetGoogle Scholar
  15. 15.
    Penrose MD, Yukich JE (2003) Weak laws of large numbers in geometric probability. Ann Appl Probab 13:277–303MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ranneby B, Jammalamadaka SR, Teterukovskiy A (2005) The maximum spacing estimation for multivariate observations J Stat Plan Inference 129:427–446MATHMathSciNetGoogle Scholar
  17. 17.
    Rényi A (1961) On measures of information and entropy. In: Proceedings of the 4th Berkeley symposium on mathematics, statistics and probability, vol 1960. University of California Press, Berkeley, pp 547–561Google Scholar
  18. 18.
    Shank N (2009) Nearest-neighbor graphs on the Cantor set. Adv Appl Probab 41:38–62MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Wade A (2007) Explicit laws of large numbers for random nearest neighbor type graphs. Adv Appl Probab 39:326–342MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Yukich JE (1998) Probability theory of classical euclidean optimization problems. Lecture notes in mathematics, vol 1675. Springer, BerlinMATHGoogle Scholar
  21. 21.
    Zhou S, Jammalamadaka SR (1993) Goodness of fit in multidimensions based on nearest neighbour distances. J Nonparametr Stat 2:271–284MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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