  Finite dimensional (FD) models, i.e., deterministic functions of time/space
and finite sets of random variables, are constructed for target vector-valued
random processes/fields. They are required to have two properties. First,
standard Monte Carlo algorithms can be used to generate their samples, referred
to as FD samples. Second, under some conditions specified by several theorems,
FD samples can be used to estimate distributions of extremes and other
functionals of target random functions. Numerical illustrations involving
two-dimensional random processes and apparent properties of random
microstructures are presented to illustrate the implementation of FD models for
these stochastic problems and show that they are accurate if the conditions of
our theorems are satisfied.
