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Foundations of Stochastic Geometry and Theory of Random Sets

  • Ilya MolchanovEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2068)

Abstract

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Keywords

Lebesgue Measure Support Function Random Point Steiner Point Stochastic Geometry 
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.

References

  1. 15.
    Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York (1980)zbMATHGoogle Scholar
  2. 17.
    Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Ann. Probab. 3, 879–882 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 62.
    Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, Berlin (1984)zbMATHCrossRefGoogle Scholar
  4. 113.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes Mathematics, vol. 580. Springer, Berlin (1977)Google Scholar
  5. 246.
    Himmelberg, C.: Measurable relations. Fund. Math. 87, 53–72 (1974)MathSciNetGoogle Scholar
  6. 292.
    Kendall, D.G.: Foundations of a theory of random sets. In: Harding, E.F., Kendall, D.G. (eds.) Stochastic Geometry. Wiley, New York (1974)Google Scholar
  7. 345.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)zbMATHGoogle Scholar
  8. 363.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005)zbMATHGoogle Scholar
  9. 366.
    Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)zbMATHGoogle Scholar
  10. 372.
    Mourier, E.: L-random elements and L  ∗ -random elements in Banach spaces. In: Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 231–242. University of California Press, Berkeley (1955)Google Scholar
  11. 384.
    Norberg, T.: Existence theorems for measures on continuous posets, with applications to random set theory. Math. Scand. 64, 15–51 (1989)MathSciNetzbMATHGoogle Scholar
  12. 451.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  13. 489.
    Stoyan, D., Kendall, W., Mecke, J.: Stochastic Geometry and Its Applications, 2nd edn. Wiley, New York (1995)zbMATHGoogle Scholar
  14. 494.
    Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)zbMATHGoogle Scholar
  15. 508.
    Weil, W.: An application of the central limit theorem for Banach-space-valued random variables to the theory of random sets. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 60, 203–208 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

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