On the convergence of random functions defined by interpolation

In the paper we study sequences of random functions which are defined by some interpolation procedures for a given random function. We investigate the problem in what sense and under which conditions the sequences converge to the prescribed random function. Sufficient conditions for convergence of moment characteristics, of finite dimensional distributions and for weak convergence of distributions in spaces of continuous functions are given. The treatment of such questions is stimulated by an investigation of Monte Carlo simulation procedures for certain classes of random functions. In an appendix basic facts concerning weak convergence of probability measures in metric spaces are summarized