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 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.