Convergence of Probability Measures. Central Limit Theorem

Abstract

Many of the fundamental results in probability theory are formulated as limit theorems. Bernoulli’s law of large numbers was formulated as a limit theorem; so was the De Moivre-Laplace theorem, which can fairly be called the origin of a genuine theory of probability and, in particular, which led the way to numerous investigations that clarified the conditions for the validity of the central limit theorem. Poisson’s theorem on the approximation of the binomial distribution by the “Poisson” distribution in the case of rare events was formulated as a limit theorem. After the example of these propositions, and of results on the rapidity of convergence in the De Moivre-Laplace and Poisson theorems, it became clear that in probability it is necessary to deal with various kinds of convergence of distributions, and to establish the rapidity of convergence connected with the introduction of various “natural” measures of the distance between distributions. In the present chapter we shall discuss some general features of the convergence of probability distributions and of the distance between them. In this section we take up questions in the general theory of weak convergence of probability measures in metric spaces. The De Moivre-Laplace theorem, the progenitor of the central limit theorem, finds a natural place in this theory. From §3, it is clear that the method of characteristic functions is one of the most powerful means for proving limit theorems on the weak convergence of probability distributions in R ⁿ. In §6, we consider questions of metrizability of weak convergence. Then, in §8, we turn our attention to a different kind of convergence of distributions (stronger than weak convergence), namely convergence in variation. Proofs of the simplest results on the rapidity of convergence in the central limit theorem and Poisson’s theorem will be given in §§10 and 11.