On the Convergence, Basic Distributions and Classification of Abstract Random Variables

We consider in this paper the convergence of abstract random variables taken in the sense of definition (1). If the random variables converge in some classical sense or in the extended sense of Frechet and Doss, then they will also converge in the sense of definition (1). The classical relations between different types of convergence, generally speaking, are preserved.
Furthermore, for a single parameter system of abstract random variables the joint distributions are defined (distributions on cylinder sets) and it is shown that convergence of the abstract random variables implies convergence of the corresponding joint distributions, and for abstract random variables with values from a separable, locally compact space, the collection of joint distributions is fully defined by the probability measure on this space.
In conclusion, a classification of abstract random variables is carried out and certain special results are obtained.