Convergence theorems for random elements in convex combination spaces

Abstract

A Vitali convergence theorem is proved for subspaces of an abstract convex combination space which admits a complete separable metric. The convergence may be in that metric or, more generally, in a quasimetric satisfying weaker properties. Versions for convergence in probability and in distribution are given. As applications, we show that some dominated convergence theorems in the literature of fuzzy random variables and random compact sets can be recovered or improved, and we derive new convergence theorems in another space of sets and in a space of probability distributions.