Skip to main content
SpringerLink
Log in

Limit Theorems in Discrete Stochastic Geometry

  • Chapter
  • First Online:
Stochastic Geometry, Spatial Statistics and Random Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2068))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Affentranger, F.: Aproximación aleatoria de cuerpos convexos. Publ. Mat. 36, 85–109 (1992)

    MathSciNet  MATH  Google Scholar 

  2. 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. Baltz, A., Dubhashi, D., Srivastav, A., Tansini, L., Werth, S.: Probabilistic analysis for a vehicle routing problem. Random Struct. Algorithms 30, 206–225 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bárány, I., Fodor, F., Vigh, V.: Intrinsic volumes of inscribed random polytopes in smooth convex bodies. Adv. Appl. Probab. 42, 605–619 (2009)

    Article  Google Scholar 

  5. 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)

    MathSciNet  Google Scholar 

  6. Baryshnikov, Y., Penrose, M., Yukich, J.E.: Gaussian limits for generalized spacings. Ann. Appl. Probab. 19, 158–185 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baryshnikov, Y., Yukich, J.E.: Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 213–253 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beardwood, J., Halton, J.H., Hammersley, J.M.: The shortest path through many points. Proc. Camb. Philos. Soc. 55, 229–327 (1959)

    Article  MathSciNet  Google Scholar 

  9. Bickel, P., Yan, D.: Sparsity and the possibility of inference. Sankhyā 70, 1–23 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999)

    Google Scholar 

  11. Buchta, C.: Zufällige Polyeder - Eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis - Lecture Notes in Mathematics, vol. 1114. Springer, Berlin (1985)

    Google Scholar 

  12. 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. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (N.S.) 46, 255–308 (2009)

    Google Scholar 

  14. Chatterjee, S.: A new method of normal approximation. Ann. Probab. 36, 1584–1610 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chazal, F., Guibas, L., Oudot, S., Skraba, P.: Analysis of scalar fields over point cloud data, Preprint (2007)

    Google Scholar 

  16. Chazal, F., Oudot, S.: Towards persistence-based reconstruction in euclidean spaces. ACM Symp. Comput. Geom. 232 (2008)

    Google Scholar 

  17. Chen, L., Shao, Q.M.: Normal approximation under local dependence. Ann. Probab. 32, 1985–2028 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Costa, J., Hero III, A.: Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. Signal Process. 58, 2210–2221 (2004)

    Article  MathSciNet  Google Scholar 

  19. 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. 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. Donoho, D., Grimes, C.: Hessian eigenmaps: locally linear embedding techniques for high dimensional data. Proc. Natl. Acad. Sci. 100, 5591–5596 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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. 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. Hero, A.O., Ma, B., Michel, O., Gorman, J.: Applications of entropic spanning graphs. IEEE Signal Process. Mag. 19, 85–95 (2002)

    Article  Google Scholar 

  25. Hille, E.: Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence (1948)

    Google Scholar 

  26. Hsing, T.: On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4, 478–493 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kesten, H., Lee, S.: The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6, 495–527 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kingman, J.F.C.: Poisson Processes, Oxford Studies in Probability. Oxford University Press, London (1993)

    MATH  Google Scholar 

  29. Kirby, M.: Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns. Wiley, New York (2001)

    MATH  Google Scholar 

  30. Koo, Y., Lee, S.: Rates of convergence of means of Euclidean functionals. J. Theoret. Probab. 20, 821–841 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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. Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields. Kluwer, Dordrecht (1991)

    Book  MATH  Google Scholar 

  33. Penrose, M.: Random Geometric Graphs. Oxford University Press, London (2003)

    Book  MATH  Google Scholar 

  34. Penrose, M.D.: Gaussian limits for random geometric measures. Electron. J. Probab. 12, 989–1035 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Penrose, M.D.: Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Penrose, M.D., Wade, A.R.: Multivariate normal approximation in geometric probability. J. Stat. Theor. Pract. 2, 293–326 (2008)

    Article  MathSciNet  Google Scholar 

  37. Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11, 1005–1041 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Penrose, M.D., Yukich, J.E.: Mathematics of random growing interfaces. J. Phys. A 34, 6239–6247 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Penrose, M.D., Yukich, J.E.: Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272–301 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  40. Penrose, M.D., Yukich, J.E.: Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13, 277–303 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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. Penrose, M.D., Yukich, J.E.: Limit theory for point processes on manifolds. Ann. Appl. Probab. (2013) (to appear). ArXiv:1104.0914

    Google Scholar 

  43. Quintanilla, J., Torquato, S.: Local volume fluctuations in random media. J. Chem. Phys. 106, 2741–2751 (1997)

    Article  Google Scholar 

  44. Redmond, C.: Boundary rooted graphs and Euclidean matching algorithms. Ph.D. thesis, Department of Mathematics, Lehigh University, Bethlehem, PA (1993)

    Google Scholar 

  45. Redmond, C., Yukich, J.E.: Limit theorems and rates of convergence for subadditive Euclidean functionals. Ann. Appl. Probab. 4, 1057–1073 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Redmond, C., Yukich, J.E.: Limit theorems for Euclidean functionals with power-weighted edges. Stoch. Process. Appl. 61, 289–304 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. Reitzner, M.: Central limit theorems for random polytopes. Probab. Theor. Relat. Fields 133, 483–507 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 75–84 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  49. Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)

    Article  Google Scholar 

  50. Schneider, R.: Random approximation of convex sets. J. Microsc. 151, 211–227 (1988)

    Article  Google Scholar 

  51. 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. Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  53. 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. Schreiber, T., Penrose, M.D., Yukich, J.E.: Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys. 272, 167–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. 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)

    Article  MathSciNet  MATH  Google Scholar 

  56. 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. Seppäläinen, T., Yukich, J.E.: Large deviation principles for Euclidean functionals and other nearly additive processes. Probab. Theor. Relat. Fields 120, 309–345 (2001)

    Article  MATH  Google Scholar 

  58. Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Probab. Theor. Relat. Fields 9, 365–376 (1981)

    MathSciNet  MATH  Google Scholar 

  59. Steele, J.M.: Probability Theory and Combinatorial Optimization. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  60. Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319B–2323 (2000)

    Article  Google Scholar 

  61. Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)

    MATH  Google Scholar 

  62. 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. Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer, Berlin (1998)

    Google Scholar 

  64. Yukich, J.E.: Limit theorems for multi-dimensional random quantizers. Electron. Comm. Probab. 13, 507–517 (2008)

    MathSciNet  MATH  Google Scholar 

  65. 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) http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/97/showToc

  66. Zuyev, S.: Strong Markov Property of Poisson Processes and Slivnyak Formula. Lecture Notes in Statistics, vol. 185. Springer, Berlin (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehigh University, Bethlehem, PA, 18015, USA

    Joseph Yukich

Authors
  1. Joseph Yukich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph Yukich .

Editor information

Editors and Affiliations

  1. Helmholtzstr. 18, Ulm, 89069, Germany

    Evgeny Spodarev

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Yukich, J. (2013). Limit Theorems in Discrete Stochastic Geometry. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_8

Download citation

Publish with us

Policies and ethics

Search

Navigation