Four Finite Dimensional (FD) Surrogates for Continuous Random Processes

Numerical solutions of stochastic problems involve approximations of the random functions in their definitions by deterministic functions of time and/or space and finite numbers of random variables, referred to as finite dimensional (FD) random functions. Since the solutions of these problems are complex functionals of random functions, the resulting numerical solutions are informative only if target random functions and their FD representations have similar samples. We establish conditions under which extremes and other functionals of real-valued continuous processes X ( t ) can be estimated from samples of FD processes X d ( t ) . Four types of FD processes are developed for stationary and nonstationary processes X ( t ). Numerical examples are presented to illustrate the construction of these FD processes and assess their capabilities to characterize extremes of X ( t ).