The Convergence of Distributions of Functionals on Stochastic Processes

The first part of the paper contains a survey of the present state of research in the general problems of convergence of stochastic processes. An important place is given to theorems on weak convergence of distributions on metric spaces. Then a different approach is proposed to the study of convergence of distributions of functionals on stochastic processes, which is connected with the approximation of the trajectories of a process by some family of functions. We think of approximation in terms of the nearness of the functionals in question. Using this approach we obtain all the main results on convergence in specific function spaces that are known at present. These results are obtained in their most general form without the requirement that the limiting processes should belong to the space under discussion. New limit theorems are also obtained and among them theorems for processes with discontinuities of the second kind and others.