Molchanov Theory of Random Sets

1. Auflage 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

Stochastic geometry is a relatively new branch of mathematics. Although its predecessors such as geometric probability date back to the 18th century, the formal concept of a random set was developed in the beginning of the 1970s. Theory of Random Sets presents a state of the art treatment of the modern theory, but it does not neglect to recall and build on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference of the 1990s. The book is entirely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time, fixes terminology and notation that are often varying in the current literature to establish it as a natural part of modern probability theory, and to provide a platform for future development.

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. Appendices: Topological Spaces.- Linear Spaces.- Space of Closed Sets.- Compact Sets and the Hausdorff Metric.- Multifunctions and Continuity.- Measures and Probabilities.- Capacities.- Convex Sets.- Semigroups and Harmonic Analysis.- Regular Variation. References.- List of Notation.- Name Index.- Subject Index.

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.