Local Central Limit Theorems in Stochastic Geometry

Mathew D. Penrose (University of Bath)
Yuval Peres (Microsoft Research)


We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance. We apply this result to various quantities arising in stochastic geometry, including: size of the largest component for percolation on a box; number of components, number of edges, or number of isolated points, for random geometric graphs; covered volume for germ-grain coverage models; number of accepted points for finite-input random sequential adsorption; sum of nearest-neighbour distances for a random sample from a continuous multidimensional distribution.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2509-2544

Publication Date: December 3, 2011

DOI: 10.1214/EJP.v16-968


  1. F. Avram, D. Bertsimas. On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 (1993), 1033-1046. Math. Review 95d:60022
  2. Yu. Baryshnikov, M.D. Penrose, J.E. Yukich. Gaussian limits for generalized spacings. Ann. Appl. Probab. 19 (2009), 158--185. Math. Review 2010d:60051
  3. Yu. Baryshnikov, J.E. Yukich. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005), 213--253. Math. Review 2005j:60043
  4. E.A. Bender,. Central and local limit theorems applied to asymptotic enumeration. J. Combinatorial Theory A 15 (1973), 91--111. Math. Review 51 #11626
  5. P. J. Bickel, L. Breiman . Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 (1983), 185--214. Math. Review 85h:60027
  6. L. Breiman. Probability. SIAM, Philadelphia (1992). Math. Review 93d:60001
  7. S. Chatterjee. A new method of normal approximation. Ann. Probab. 36 (2008), 1584--1610. Math. Review 2009j:60043
  8. B. Davis, D. McDonald. An elementary proof of the local central limit theorem. J. Theoret. Probab. 8 (1995), 693--701. Math. Review 96j:60037
  9. R. Durrett. Probability: Theory and Examples. 2nd Edition, Wadsworth, Belmont, CA. (1996) Math. Review 98m:60001
  10. D. Evans, A.J. Jones. A proof of the gamma test. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), 2759--2799. Math. Review 2003i:62081
  11. W. Feller. An Introduction to Probability Theory and its Applications. Vol. II. John Wiley & Sons, New York (1966). Math. Review 35 #1048
  12. G. Grimmett, J.M. Marstrand. The supercritical phase of percolation is well behaved. Proc. Royal Soc. London A 430 (1990), 439--457. Math. Review 91m:60186
  13. L. Heinrich, I.S. Molchanov. Central limit theorem for a class of random measures associated with germ-grain models. Adv. Appl. Probab. 31 (1999), 283--314. Math. Review 2001a:60013
  14. N. Henze. A multivariate two-sample test based on the number of nearest neighbor type coincidences. Ann. Statist. 16 (1988), 772--783. Math. Review 89g:62083
  15. J.F.C. Kingman. Poisson Processes. Oxford Studies in Probability 3. (1993) Oxford University Press. Math. Review 94a:60052
  16. E. Levina, P.J. Bickel. Maximum likelihood estimation of intrinsic dimension. In Advances in NIPS, 17, Eds. L. K. Saul, Y. Weiss, L. Bottou (2005). Math. Review number not available
  17. N. Leonenko, L. Pronzato, V. Savani. A class of Renyi information estimators for multidimensional densities. Ann. Statist. 36(2008), 2153--2182. Math. Review 2010c:94013
  18. M. Penrose. Random Geometric Graphs. Oxford Studies in Probability 5. (2003) Oxford University Press. Math. Review 2005j:60003
  19. M.D. Penrose,. A central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Probab. 29 (2001), 1515--1546. Math. Review 2002m:60040
  20. M.D. Penrose. Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007), 989--1035. Math. Review
  21. M.D. Penrose, J.E. Yukich. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001), 1005--1041. Math. Review 2002k:60068
  22. M.D. Penrose, J.E. Yukich. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002), 272--301. Math. Review 2003c:60046
  23. M.D. Penrose, J.E. Yukich. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003), 277--303. Math. Review 2009b:60039
  24. M.D. Penrose, J.E. Yukich (2011). Limit theory for point processes in manifolds. Preprint, ArXiv:1104.0914. Math. Review number not available.
  25. T. Schreiber, M.D. Penrose, J.E. Yukich . Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys. 272 (2007), 167--183. Math. Review 2008k:60056
  26. N.N. Vakhania. Elementary proof of Polya's characterization theorem and of the necessity of second moment in the CLT. Theory Probab. Appl. 38 (1993), 166--168. Math. Review 96a:60015

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.