On Convergence of Random Variables and Distributions

Abstract

In this chapter, several different types of convergence used in Probability Theory are defined and relationships between them are elucidated. Section 6.1 deals with convergence in probability and convergence with probability one (the almost sure convergence), presenting some criteria for them and, in particular, discussing the concept of Cauchy sequences (in probability and almost surely). Then the continuity theorem is established (convergence of functions of random variables) and the concept of uniform integrability is introduced and discussed, together with its consequences (in particular, for convergence in mean of suitable orders). Section 6.2 contains an extensive discussion of weak convergence of distributions. The chapter ends with Sect. 6.3 presenting criteria for weak convergence of distributions, including the concept of distribution determining classes of functions and that of tightness.