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.