From Convergence of Functions to Convergence of Stochastic Processes. on Skorokhod's Sequential Approach to Convergence in Distribution

Motivated by original Skorokhod's ideas, a new topology has been deened on the space P(X) of tight probability distributions on a topological space (X;). The only topological assumption imposed on (X;) is that some countable family of continuous functions separates points of X. This new sequential topology, deened by means of a variant of the a.s. Skorokhod representation, is quite operational and from the point of view of nonmetric spaces proves to be more satisfactory than the weak topology. In particular, in this topology the direct Prohorov's theorem preserves its distinguished position within the theory and the converse Prohorov's theorem is quite natural and holds in many spaces. The topology coincides with the usual topology of weak convergence when (X;) is a metric space or a space of distributions (like S 0 or D 0).