Topics in Weak Convergence of Probability Measures

Milan Merkle

The weak convergence is a very powerful tool in Probability Theory, partly due to its comparative simplicity and partly due to its natural behavior in some typical problems.The concept of weak convergence is so well established in Probability Theory that hardly any textbook even mention its topological heritage. It, indeed, is not too important in many applications, but a complete grasp of the definition of the weak convergence is not possible without understanding its rationale. The first part of this paper (Sections 2 and 3) is an introduction to weak convergence of probability measures from the topological point of view. Since the set of probability measures is not closed under weak convergence (as we shall see, the limit of a net of probability measures need not be a probability measure), for a full understanding of the complete concept, one has to investigate a wider structure, which turns out to be the set of all finitely additive Radon measures. In this context we present results concerning the Baire field and sigma field, which are usually omitted when discussing probability measures. In Section 4 we consider weak convergence of probability measures and present classical results regarding metrics of weak convergence. In Section 5 we show that the set of probability measures is not closed and effectively show the existence of a finitely, but not countably additive measure in the closure of the set of probability measures. Section 6 deals with the famous Prohorov's theorem on metric spaces. In Section 7 we consider weak convergence of probability measures on Hilbert spaces. Here we observe a separable Hilbert space equipped with weak and strong topology and in both cases we give necessary and sufficient conditions for relative compactness of a set of probability measures.