Foundations of stochastic geometry and theory of random sets

Molchanov, Ilya (2013). Foundations of stochastic geometry and theory of random sets. In: Spodarev, Evgeny (ed.) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics: Vol. 2068 (pp. 1-20). Berlin: Springer 10.1007/978-3-642-33305-7_1

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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.

Item Type:	Book Section (Book Chapter)
Division/Institute:	08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science
UniBE Contributor:	Molchanov, Ilya
Subjects:	500 Science > 510 Mathematics
ISSN:	0075-8434
ISBN:	978-3-642-33305-7
Series:	Lecture Notes in Mathematics
Publisher:	Springer
Language:	English
Submitter:	Lutz Dümbgen
Date Deposited:	01 Apr 2014 03:16
Last Modified:	05 Dec 2022 14:28
Publisher DOI:	10.1007/978-3-642-33305-7_1
URI:	https://boris.unibe.ch/id/eprint/41516