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.
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Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York (1980)
Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Ann. Probab. 3, 879–882 (1975)
Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, Berlin (1984)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes Mathematics, vol. 580. Springer, Berlin (1977)
Himmelberg, C.: Measurable relations. Fund. Math. 87, 53–72 (1974)
Kendall, D.G.: Foundations of a theory of random sets. In: Harding, E.F., Kendall, D.G. (eds.) Stochastic Geometry. Wiley, New York (1974)
Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)
Molchanov, I.: Theory of Random Sets. Springer, London (2005)
Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)
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)
Norberg, T.: Existence theorems for measures on continuous posets, with applications to random set theory. Math. Scand. 64, 15–51 (1989)
Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)
Stoyan, D., Kendall, W., Mecke, J.: Stochastic Geometry and Its Applications, 2nd edn. Wiley, New York (1995)
Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)
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)
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Molchanov, I. (2013). Foundations of Stochastic Geometry and Theory of Random Sets. 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_1
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DOI: https://doi.org/10.1007/978-3-642-33305-7_1
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