Convergence in Distribution on the Real Line

Abstract

In this chapter, we introduce a concept of convergence for sequences of distributions on ℝ and ℝ̄. This ‘convergence in distribution’ gives us a rigorous way to express the idea that two distributions are close to each other. For instance, we show in an example that a Poisson distribution can be approximated arbitrarily closely by binomial distributions. An important result, the Continuity Theorem, gives a criterion for the convergence of a sequence of distributions in terms of the corresponding sequence of characteristic functions. There are several other useful criteria as well, which are collected together in a result known as the Portmanteau Theorem. Although the most important applications of convergence in distribution will be found in later chapters, some are included here, including an introduction to the theory of ‘extreme values’, a discussion of the effects that ‘scaling’ and ‘centering’ have on sequences of distributions, and characterizations of moment generating functions and characteristic functions.