The book concerns limit theorems and laws of large numbers for scaled unionsof independent identically distributed random sets. These results generalizewell-known facts from the theory of extreme values. Limiting distributions (called union-stable) are characterized and found explicitly for many examples of random closed sets. The speed of convergence in the limit theorems for unions is estimated by means of the probability metrics method.It includes the evaluation of distances between distributions of random sets constructed similarly to the well-known distances between distributions of random variables. The techniques include regularly varying functions, topological properties of the space of closed sets, Choquet capacities, convex analysis and multivalued functions. Moreover, the concept of regular variation is elaborated for multivalued (set-valued) functions. Applications of the limit theorems to simulation of random sets, statistical tests, polygonal approximations of compacts, limit theorems for pointwise maxima of random functions are considered. Several open problems are mentioned. Addressed primarily to researchers in the theory of random sets, stochastic geometry and extreme value theory, the book will also be of interest to applied mathematicians working on applications of extremal processes and their spatial counterparts. The book is self-contained, and no familiarity with the theory of random sets is assumed.

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Bibliographic Information

Book Title: Limit Theorems for Unions of Random Closed Sets
Authors: Ilya S. Molchanov
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/BFb0073527
Publisher: Springer Berlin, Heidelberg
eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag Berlin Heidelberg 1993
Softcover ISBN: 978-3-540-57393-7Published: 29 November 1993
eBook ISBN: 978-3-540-48111-9Published: 15 November 2006
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: X, 158
Topics: Probability Theory and Stochastic Processes

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Table of contents (8 chapters)

Front Matter

Back Matter

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