Advertisement

Journal of Statistical Theory and Practice

, Volume 2, Issue 2, pp 293–326 | Cite as

Multivariate Normal Approximation in Geometric Probability

Article

Abstract

Consider a measure μλ = Σxξxδx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in ℝd are asymptotically independent normals as λ → ℞ here we give an O(λ−1/(2d+ε)) bound on the rate of convergence, and also a new criterion for the limiting normals to be non-degenerate. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

Key-words

Multivariate normal approximation geometric probability stabilization central limit theorem Stein’s method nearest-neighbour graph 

AMS Subject Classification

60D05 60F05 60G57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M., Stegun, I. A. (eds.), 1965. Handbook of Mathematical Functions, National Bureau of Standards. Applied Mathematics Series 55. U.S. Government Printing Office, Washington D.C.Google Scholar
  2. Avram, F., Bertsimas, D., 1993. On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, 1033–1046.MathSciNetCrossRefGoogle Scholar
  3. Baldi, P., Rinott, Y., 1989. On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17, 1646–1650.MathSciNetCrossRefGoogle Scholar
  4. Baryshnikov, Yu., Penrose, M. D., Yukich, J. E., 2008. Gaussian limits for generalized spacings. Ann. Appl. Probab. to appear.Google Scholar
  5. Baryshnikov, Yu., Yukich, J. E., 2005. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 213–253.MathSciNetCrossRefGoogle Scholar
  6. Chen, L. H. Y., Shao, Q.-M., 2004. Normal approximation under local dependence. Ann. Probab. 32, 1985–2028.MathSciNetCrossRefGoogle Scholar
  7. Durrett, R., 2004. Probability: Theory and Examples, 3rd edition, Duxbury/Brooks Cole.MATHGoogle Scholar
  8. Götze, F., 1991. On the rate of convergence in the multivariate CLT. Ann. Probab. 19, 724–739.MathSciNetCrossRefGoogle Scholar
  9. Goldstein, L., Rinott, Y., 1996. On multivariate normal approximations by Stein’s method. J. Appl. Probab. 33, 1–17.MathSciNetCrossRefGoogle Scholar
  10. Kesten, H., Lee, S., 1996. The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6, 495–527.MathSciNetCrossRefGoogle Scholar
  11. Kingman, J. F. C., 1993. Poisson Processes, Oxford Studies in Probability 3, Clarendon Press, Oxford.Google Scholar
  12. Penrose, M., 2003. Random Geometric Graphs, Oxford Studies in Probability 6, Clarendon Press, Oxford.Google Scholar
  13. Penrose, M. D., 2005a. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33, 1945–1991.MathSciNetCrossRefGoogle Scholar
  14. Penrose, M. D., 2005b. Convergence of random measures in geometric probability. Preprint available from http://arxiv.org/abs/math.PR/0508464.Google Scholar
  15. Penrose, M. D., 2007a. Gaussian limits for random geometric measures. Electron. J. Probab. 12 (paper 35), 989–1035.MathSciNetCrossRefGoogle Scholar
  16. Penrose, M. D., 2007b. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.MathSciNetCrossRefGoogle Scholar
  17. Penrose, M. D., Wade, A. R., 2008. Limit theory for the random on-line nearest-neighbour graph. Random Structures Algorithms 32, 125–156.MathSciNetCrossRefGoogle Scholar
  18. Penrose, M. D., Yukich, J. E., 2001. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11, 1005–1041.MathSciNetMATHGoogle Scholar
  19. Penrose, M. D., Yukich, J. E., 2002. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272–301.MathSciNetCrossRefGoogle Scholar
  20. Penrose, M. D., Yukich, J. E., 2003. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13, 277–303.MathSciNetCrossRefGoogle Scholar
  21. Penrose, M. D., Yukich, J. E., 2005. Normal approximation in geometric probability. In Stein’s Method and Applications, eds. Louis H. Y. Chen, A. D. Barbour, Lecture Notes Series, Institute for Mathematical Sciences, Vol. 5, World Scientific, Singapore.CrossRefGoogle Scholar
  22. Rinott, Y., Rotar, V., 1996. A multivariate CLT for local dependence with n−1/2logn rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56, 333–350.MathSciNetCrossRefGoogle Scholar
  23. Stein, C., 1972. Approximate Computation of Expectations, IMS, Hayward, CA.MATHGoogle Scholar
  24. Wade, A. R., 2007. Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. Appl. Probab. 39, 326–342.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathEngland
  2. 2.Department of MathematicsUniversity of Bristol, University WalkBristolEngland

Personalised recommendations