TABLE OF CONTENTS The three types of convergence: convergence in probability; convergence in distribution; and almost-sure convergence are related to each other and to convergence in mean of order p as shown in Figure 14.1. In particular, convergence in mean of order p does not imply almost-sure convergence, and almost-sure convergence does not imply convergence in mean of order p for any p. The Exam preparation section at the end of the chapter contains a summary of important facts about the different kinds of convergence.
Section 14.1 introduces the notion of convergence in probability. Convergence in probability was important in Chapter 6 on parameter estimation and confidence intervals, where it justifies various statistical procedures that are used to estimate unknown parameters.
Section 14.2 introduces the notion of convergence in distribution. Convergence in distribution is often used to approximate probabilities that are hard to calculate exactly. Suppose that X n is a random variable whose cumulative distribution function 
is hard to compute. But suppose that for large n, 
, where F X( x) is a cdf that is easy to compute. Loosely speaking, when 
, we say that X n converges in distribution to X. In this case, we can approximate by F X( x) if n is large enough. When the central limit theorem applies, F X is the normal cdf with mean zero and variance...
