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Random Closed Sets

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Space, Structure and Randomness

Part of the book series: Lecture Notes in Statistics ((LNS,volume 183))

Abstract

Concepts and results involving random sets appeared in probabilistic and statistical literature long time ago. The origin of the modern concept of a random set goes as far back as the seminal book by A.N. Kolmogorov [22] (first published in 1933) where he laid out the foundations of probability theory. He wrote [22, p. 46]

Let G be a measurable region of the plane whose shape depends on chance; in other words let us assign to every elementary event ξ of a field of probability a definite measurable plane region G. In modern terminology, G is said to be a random set, which is not necessarily closed, see [37, Sec. 2.5]. It should be noted also that even before 1933 statisticians worked with confidence regions that can be naturally described as random sets.

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Authors and Affiliations

  1. Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH 3012, Bern, Switzerland

    I. Molchanov

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  1. I. Molchanov
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Editors and Affiliations

  1. Centre de Morphologie Mathématique, Ecole des Mines de Paris, 35 rue Saint Honoré, 77305, Fontainebleau, France

    Michel Bilodeau  & Fernand Meyer  & 

  2. Direction des recherches, Ecole des Mines de Paris, 60 Boulevard Saint Michel, 75272, Paris Cedex 06, France

    Michel Schmitt

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Molchanov, I. (2005). Random Closed Sets. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds) Space, Structure and Randomness. Lecture Notes in Statistics, vol 183. Springer, New York, NY. https://doi.org/10.1007/0-387-29115-6_7

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