Molchanov Theory of Random Sets

2005
ISBN: 978-1-84628-150-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 488 Seiten, eBook

Reihe: Probability and Its Applications

This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notes

Interdisciplinary connections and applications of random sets are emphasized throughout the book

An extensive bibliography in the book is available on the Web at http://liinwww.ira.uka.de/bibliography/math/random.closed.sets.html, and is accompanied by a search engine

Zielgruppe

Research

Autoren/Hrsg.

Molchanov, Ilya

Inhaltsverzeichnis

Random Closed Sets and Capacity Functionals.- Expectations of Random Sets.- Minkowski Addition.- Unions of Random Sets.- Random Sets and Random Functions.

Leseproben

2 Expectations of Random Sets (p.145)

1 The selection expectation

The space F of closed sets (and also the space K of compact sets) is non-linear, so that conventional concepts of expectations in linear spaces are not directly applicable for random closed (or compact) sets. Sets have different features (that often are dif.cult to express numerically) and particular definitions of expectations highlight various features important in the chosen context.

To explain that an expectation of a random closed (or compact) set is not straightforward to de.ne, consider a random closed set X which equals [0, 1] with probability 1/2 and otherwise is {0, 1}. For another example, let X be a triangle with probability 1/2 and a disk otherwise. A "reasonable" expectation in either example is not easy to de.ne. Strictly speaking, the de.nition of the expectation depends on what the objective is, which features of random sets are important to average and which are possible to neglect.

This section deals with the selection expectation (also called the Aumann expectation), which is the best investigated concept of expectation for random sets. Since many results can be naturally formulated for random closed sets in Banach spaces, we assume that E is a separable Banach space unless stated otherwise. Special features inherent to expectations of random closed sets in Rd will be highlighted throughout. To avoid unnecessary complications, it is always assumed that all random closed sets are almost surely non-empty.

1.1 Integrable selections

The key idea in the de.nition of the selection expectation is to represent a random closed set as a family of its integrable selections. The concept of a selection of a random closed set was introduced in De.nition 1.2.2.While properties of selections discussed in Section 1.2.1 can be formulated without assuming a linear structure on E, now we discuss further features of random selections with the key issue being their integrability.

Über Autor(innen)

Ilya Molchanov is Professor of Probability Theory in the Department of Mathematical Statistics and Actuarial Science at the University of Berne, Switzerland.