Limit Theorems for Unions of Random Closed Sets
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 ch
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Ilya S. Molchanov
Limit Theorems for Unions of Random Closed Sets
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Author Ilya S. Molchanov Department of Mathematics Kiev Technological Institute of the Food Industry Vladimirskaya 68 252017 Kiev, Ukraine and FB Mathematik TU Bergakademie Freiberg Bemhard-v.-Cotta-Str. 2 D-09596 Freiberg, Germany
Mathematics Subject Classification (1991): 60-02, 60D05, 60055, 60G70, 26E25, 28B20,52A22 ISBN 3-540-57393-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57393-3 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper
PREFACE The theory of geometrical probability is, certainly, one of the oldest branches of probability theory. It deals with probability distributions on spaces of geometrical objects (points, lines, planes, triangles, sets etc.) and the corresponding random elements, see Ambartzumian (1990). The notion of a random closed set was introduced by Kendall (1974) and Matheron (1975). Since their studies the concept of probability was defined in a satisfactory manner from the point of view of probability measure on a space of closed sets. Although a random closed set is a special case of general random elements, random sets have special properties due to the topological structure of the space of closed sets and specific features of set-theoretic operations. Therefore, well-known theorems of classical probability theory gain new meanings and features within the framework of the theory of random sets. The role and place of limit theorems in probability theory can scarcely be exaggerated. Many important distributions appear as limiting ones with respect to various operations. It is of great interest to derive limit theorems for random sets with respect to set-theoretic operations such as union, intersection or Minkowski (element-wise) addition. It should be noted that limit theorems for random vectors will naturally follow from limit theorems for random sets, since a random vector can be considered to be a single-point random set. On the other hand, limit theorems for random sets gain new features as long as we deal with shapes of limiting random sets and summands. The limit theorems for random sets have been investigated mostly for the Minkowski addition. The properties of this operation imply that the limiting distribution corre
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