Limit Theorems for Unions of Random Closed Sets

Ilya S. Molchanov (auth.)3540573933, 9783540573937, 0387573933

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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Table of contents :
Distributions of random closed sets….Pages 1-14
Survey on stability of random sets and limit theorems for Minkowski addition….Pages 15-27
Infinite divisibility and stability of random sets with respect to unions….Pages 29-44
Limit theorems for normalized unions of random closed sets….Pages 45-65
Almost sure convergence of unions of random closed sets….Pages 67-84
Multivalued regularly varying functions and their applications to limit theorems for unions of random sets….Pages 85-99
Probability metrics in the space of random sets distributions….Pages 101-121
Applications of limit theorems….Pages 123-145