Limit Theorems for Unions of Random Closed Sets

The book concerns limit theorems and laws of large numbersfor scaled unionsof independent identically distributedrandom sets. These results generalizewell-known facts fromthe theory of extreme values. Limiting distributions (calledunion-stable) are characterized and found explicitly formany examples of random closed sets. The speed ofconvergence in the limit theorems for unions is estimated bymeans of the probability metrics method.It includes theevaluation of distances between distributions of randomsets constructed similarly to the well-known distancesbetween distributions of random variables. The techniquesinclude regularly varying functions, topological propertiesof the space of closed sets, Choquet capacities, convexanalysis and multivalued functions.Moreover, the concept of regular variation is elaborated formultivalued (set-valued) functions. Applications of thelimit theorems to simulation of random sets, statisticaltests, polygonal approximations of compacts, limit theoremsfor pointwise maxima of random functions are considered.Several open problems are mentioned.Addressed primarily to researchers in the theory of randomsets, stochastic geometry and extreme value theory, the bookwill also be of interest to applied mathematicians workingon applications of extremal processes and their spatialcounterparts. The book is self-contained, and no familiaritywith the theory of random sets is assumed.