A Characterization of the Weak Convergence of Measures

Abstract

In this paper we shall investigate the so-called weak convergence of measures. Although the origin of the concept of the weak convergence of measures is a probabilistic one, the concept itself is purely measure-theoretical, and should be, therefore, treated by measure-theoretical methods. In Probability Theory the notion of the weak convergence of measures first appeared in Central Limit Problem. Its full importance, however, has been recognized only recently. It is now known as Donsker's Invariance Principle. In this paper we shall follow Prohorov's approach, as presented in [1]. The list of all necessary definitions and results is given in the Introduction. We shall give some conditions for the weak convergence of measures in separable and complete metric spaces, which are expressed in terms of convergence of measures generated in finite dimensional Euclidean spaces. The last convergence can be treated by standard mathematical tools, like the Theory of Fourier Transformations. It should be noted that our theorems concerning the convergence of measures in separable complete metric spaces remain valid if we omit the assumption of completeness. The proofs will remain essentially unchanged; only instead of dealing with compact sets, we should deal with totally bounded closed sets. The theorems given in Section 4 are of interest for the Theory of Stochastic Processes, since they give the conditions for the weak convergence of measures in the functional spaces $D\lbrack 0, 1\rbrack$ and $C\lbrack 0, 1\rbrack$, and to a large class of stochastic processes there correspond measures generated in space $C\lbrack 0, 1\rbrack$, and these measures are usually given in terms of $\mu^{t_1, \cdots, t_m}$, i.e. in terms of finite dimensional distribution functions of the process.