Limit Theorems in Discrete Stochastic Geometry

  • Joseph YukichEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2068)


We survey two general methods for establishing limit theorems for functionals in discrete stochastic geometry. The functionals are linear statistics with the general representation \(\sum _{x\in \mathcal{X}}\xi (x,\mathcal{X})\), where \(\mathcal{X}\) is finite and where the interactions of x with respect to \(\mathcal{X}\), given by \(\xi (x,\mathcal{X})\), are spatially correlated. We focus on subadditive methods and stabilization methods as a way to obtain weak laws of large numbers, variance asymptotics, and central limit theorems for normalized and re-scaled versions of \(\sum _{i=1}^{n}\xi (\eta _{i},\{\eta _{j}\}_{j=1}^{n})\), where η j , j ≥ 1, are i.i.d. random variables. The general theory is applied to deduce the limit theory for functionals arising in Euclidean combinatorial optimization, convex hulls of i.i.d. samples, random sequential packing, and dimension estimation.


Central Limit Theorem Minimum Span Tree Variance Asymptotics Poisson Point Process Cumulant Measure 
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.


  1. 3.
    Affentranger, F.: Aproximación aleatoria de cuerpos convexos. Publ. Mat. 36, 85–109 (1992)MathSciNetzbMATHGoogle Scholar
  2. 12.
    Anandkumar, A., Yukich, J.E., Tong, L., Swami, A.: Energy scaling laws for distributed inference in random networks. IEEE J. Sel. Area. Comm., Issue on Stochastic Geometry and Random Graphs for Wireless Networks 27, 1203–1217 (2009)Google Scholar
  3. 39.
    Baltz, A., Dubhashi, D., Srivastav, A., Tansini, L., Werth, S.: Probabilistic analysis for a vehicle routing problem. Random Struct. Algorithms 30, 206–225 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 46.
    Bárány, I., Fodor, F., Vigh, V.: Intrinsic volumes of inscribed random polytopes in smooth convex bodies. Adv. Appl. Probab. 42, 605–619 (2009)CrossRefGoogle Scholar
  5. 55.
    Baryshnikov, Y., Eichelsbacher, P., Schreiber, T., Yukich, J.E.: Moderate deviations for some point measures in geometric probability. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 44, 442–446 (2008)MathSciNetGoogle Scholar
  6. 56.
    Baryshnikov, Y., Penrose, M., Yukich, J.E.: Gaussian limits for generalized spacings. Ann. Appl. Probab. 19, 158–185 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 57.
    Baryshnikov, Y., Yukich, J.E.: Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 213–253 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 61.
    Beardwood, J., Halton, J.H., Hammersley, J.M.: The shortest path through many points. Proc. Camb. Philos. Soc. 55, 229–327 (1959)MathSciNetCrossRefGoogle Scholar
  9. 67.
    Bickel, P., Yan, D.: Sparsity and the possibility of inference. Sankhyā 70, 1–23 (2008)MathSciNetzbMATHGoogle Scholar
  10. 69.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999)Google Scholar
  11. 88.
    Buchta, C.: Zufällige Polyeder - Eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis - Lecture Notes in Mathematics, vol. 1114. Springer, Berlin (1985)Google Scholar
  12. 111.
    Calka, P., Schreiber, T., Yukich, J.E.: Brownian limits, local limits and variance asymptotics for convex hulls in the ball. Ann. Probab. (2012) (to appear)Google Scholar
  13. 112.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (N.S.) 46, 255–308 (2009)Google Scholar
  14. 116.
    Chatterjee, S.: A new method of normal approximation. Ann. Probab. 36, 1584–1610 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 118.
    Chazal, F., Guibas, L., Oudot, S., Skraba, P.: Analysis of scalar fields over point cloud data, Preprint (2007)Google Scholar
  16. 119.
    Chazal, F., Oudot, S.: Towards persistence-based reconstruction in euclidean spaces. ACM Symp. Comput. Geom. 232 (2008)Google Scholar
  17. 120.
    Chen, L., Shao, Q.M.: Normal approximation under local dependence. Ann. Probab. 32, 1985–2028 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 130.
    Costa, J., Hero III, A.: Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. Signal Process. 58, 2210–2221 (2004)MathSciNetCrossRefGoogle Scholar
  19. 131.
    Costa, J., Hero III, A.: Determining intrinsic dimension and entropy of high-dimensional shape spaces. In: Krim, H., Yezzi, A. (eds.) Statistics and Analysis of Shapes. Birkhäuser, Basel (2006)Google Scholar
  20. 140.
    Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Vol. I and II. Probab. Appl. (New York). Springer, New York (2003/2008)Google Scholar
  21. 156.
    Donoho, D., Grimes, C.: Hessian eigenmaps: locally linear embedding techniques for high dimensional data. Proc. Natl. Acad. Sci. 100, 5591–5596 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 163.
    Dvoretzky, A., Robbins, H.: On the “parking” problem. MTA Mat. Kut. Int. Kől. (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 9, 209–225 (1964)Google Scholar
  23. 207.
    Gruber, P.M.: Comparisons of best and random approximations of convex bodies by polytopes. Rend. Circ. Mat. Palermo (2) Suppl. 50, 189–216 (1997)Google Scholar
  24. 242.
    Hero, A.O., Ma, B., Michel, O., Gorman, J.: Applications of entropic spanning graphs. IEEE Signal Process. Mag. 19, 85–95 (2002)CrossRefGoogle Scholar
  25. 245.
    Hille, E.: Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence (1948)Google Scholar
  26. 248.
    Hsing, T.: On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4, 478–493 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 295.
    Kesten, H., Lee, S.: The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6, 495–527 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 298.
    Kingman, J.F.C.: Poisson Processes, Oxford Studies in Probability. Oxford University Press, London (1993)zbMATHGoogle Scholar
  29. 299.
    Kirby, M.: Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns. Wiley, New York (2001)zbMATHGoogle Scholar
  30. 309.
    Koo, Y., Lee, S.: Rates of convergence of means of Euclidean functionals. J. Theoret. Probab. 20, 821–841 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 328.
    Levina, E., Bickel, P.J.: Maximum likelihood estimation of intrinsic dimension. In: Saul, L.K., Weiss, Y., Bottou, L. (eds.) Advances in NIPS, vol. 17 (2005)Google Scholar
  32. 341.
    Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields. Kluwer, Dordrecht (1991)CrossRefzbMATHGoogle Scholar
  33. 394.
    Penrose, M.: Random Geometric Graphs. Oxford University Press, London (2003)CrossRefzbMATHGoogle Scholar
  34. 395.
    Penrose, M.D.: Gaussian limits for random geometric measures. Electron. J. Probab. 12, 989–1035 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 396.
    Penrose, M.D.: Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 397.
    Penrose, M.D., Wade, A.R.: Multivariate normal approximation in geometric probability. J. Stat. Theor. Pract. 2, 293–326 (2008)MathSciNetCrossRefGoogle Scholar
  37. 398.
    Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11, 1005–1041 (2001)MathSciNetzbMATHGoogle Scholar
  38. 399.
    Penrose, M.D., Yukich, J.E.: Mathematics of random growing interfaces. J. Phys. A 34, 6239–6247 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 400.
    Penrose, M.D., Yukich, J.E.: Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272–301 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 401.
    Penrose, M.D., Yukich, J.E.: Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13, 277–303 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 402.
    Penrose, M.D., Yukich, J.E.: Normal approximation in geometric probability. In: Barbour, A.D., Chen, L.H.Y. (eds.) Stein’s Method and Applications. Lecture Note Series, vol. 5. Institute for Mathematical Sciences, National University of Singapore, 37–58 (2005)Google Scholar
  42. 403.
    Penrose, M.D., Yukich, J.E.: Limit theory for point processes on manifolds. Ann. Appl. Probab. (2013) (to appear). ArXiv:1104.0914Google Scholar
  43. 410.
    Quintanilla, J., Torquato, S.: Local volume fluctuations in random media. J. Chem. Phys. 106, 2741–2751 (1997)CrossRefGoogle Scholar
  44. 415.
    Redmond, C.: Boundary rooted graphs and Euclidean matching algorithms. Ph.D. thesis, Department of Mathematics, Lehigh University, Bethlehem, PA (1993)Google Scholar
  45. 416.
    Redmond, C., Yukich, J.E.: Limit theorems and rates of convergence for subadditive Euclidean functionals. Ann. Appl. Probab. 4, 1057–1073 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 417.
    Redmond, C., Yukich, J.E.: Limit theorems for Euclidean functionals with power-weighted edges. Stoch. Process. Appl. 61, 289–304 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 419.
    Reitzner, M.: Central limit theorems for random polytopes. Probab. Theor. Relat. Fields 133, 483–507 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 421.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 75–84 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 427.
    Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  50. 444.
    Schneider, R.: Random approximation of convex sets. J. Microsc. 151, 211–227 (1988)CrossRefGoogle Scholar
  51. 446.
    Schneider, R.: Discrete aspects of stochastic geometry. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (1997)Google Scholar
  52. 451.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  53. 457.
    Schreiber, T.: Limit theorems in stochastic geometry. In: Kendall, W.S., Molchanov. I. (eds.) New Perspectives in Stochastic Geometry. Oxford University Press, London (2010)Google Scholar
  54. 458.
    Schreiber, T., Penrose, M.D., Yukich, J.E.: Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys. 272, 167–183 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 459.
    Schreiber, T., Yukich, J.E.: Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab. 36, 363–396 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 460.
    Schreiber, T., Yukich, J.E.: Limit theorems for geometric functionals of Gibbs point processes. Ann. Inst. H. Poincaré Probab. Stat. (2012, to appear)Google Scholar
  57. 463.
    Seppäläinen, T., Yukich, J.E.: Large deviation principles for Euclidean functionals and other nearly additive processes. Probab. Theor. Relat. Fields 120, 309–345 (2001)CrossRefzbMATHGoogle Scholar
  58. 481.
    Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Probab. Theor. Relat. Fields 9, 365–376 (1981)MathSciNetzbMATHGoogle Scholar
  59. 482.
    Steele, J.M.: Probability Theory and Combinatorial Optimization. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  60. 492.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319B–2323 (2000)CrossRefGoogle Scholar
  61. 494.
    Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)zbMATHGoogle Scholar
  62. 513.
    Weil, W., Wieacker, J.A.: Stochastic geometry. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, Vol. B. North-Holland, Amsterdam (1993)Google Scholar
  63. 524.
    Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer, Berlin (1998)Google Scholar
  64. 525.
    Yukich, J.E.: Limit theorems for multi-dimensional random quantizers. Electron. Comm. Probab. 13, 507–517 (2008)MathSciNetzbMATHGoogle Scholar
  65. 526.
    Yukich, J.E.: Point process stabilization methods and dimension estimation. Proceedings of Fifth Colloquium of Mathematics and Computer Science. Discrete Math. Theor. Comput. Sci. 59–70 (2008)
  66. 532.
    Zuyev, S.: Strong Markov Property of Poisson Processes and Slivnyak Formula. Lecture Notes in Statistics, vol. 185. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehigh UniversityBethlehemUSA

Personalised recommendations