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Limit theorems for unions of random closed sets. (English) Zbl 0790.60015

Lecture Notes in Mathematics. 1561. Berlin: Springer-Verlag. x, 157 p. DM 42.00; öS 327.60; sFr 46.50/pbk (1993).
The monograph is the first to offer a self-contained presentation of the new developments on limit theory for unions of random closed sets. The results included into the book are generalizations of the max-stability concept for i.i.d. random vectors and regularly varying functions at a high level. A large number of examples illustrates the main ideas and possibilities for applications.
In Chapter 1 the basic notions and known results are introduced, e.g. random set distributions, their weak convergence, capacity functionals. Chapter 2 is a brief account on limit results for Minkowski addition. The infinite divisibility, union-stability and convex-stability of random sets are characterized in Chapter 3 in terms of their capacity and inclusion functionals, respectively. Here one can find a version of the convergence of type theorem for homogeneous at infinity random sets. In Chapter 4 necessary and sufficient conditions are discussed for convergence of normalized unions and convex hulls of random sets. The explicit form of the limiting capacity, resp. inclusion functional is found for special random sets. Several open problems are listed. Chapter 5 is devoted to the strong law of large numbers for unions of random sets. In Chapter 6 limit theorems for unions are reformulated by using integrals on multivalued regularly varying functions in the expression of the limiting capacity functional. In Chapter 7 the rate of convergence for random set distributions demonstrates the power of the probability metrics method, elaborated by V. M. Zolotarev (1986). The last Chapter 8 contains a wide range of applications of the limit theory, e.g. to simulation of stable random sets, estimation of tail probabilities for volumes of random samples, limit results for random sets generated by graphs of random functions, convergence of random processes generated by approximations of convex compact sets. The monograph ends with a list of references containing 98 items. It will be of interest mainly for researchers in the theory of random sets, extremal processes, regular variation, stochastic geometry and can serve also as a very useful source for graduate university courses.
Reviewer: E.Pancheva (Sofia)

MSC:

60D05 Geometric probability and stochastic geometry
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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