TABLE OF CONTENTS As mentioned at the beginning of Chapter 1, limit theorems are the foundation of the success of Kolmogorov s axiomatic theory of probability. In this chapter and the next, we focus on four different notions of convergence and their implications. The four types of convergence are, in the order to be studied:
convergence in mean of order p;
convergence in probability;
convergence in distribution; and
almost sure convergence.
When we say X n converges to X, we usually understand this intuitively as what is known as almost-sure convergence. However, when we want to talk about moments, say E[ 
] ? E[ X 2], we need to exploit results based on convergence in mean of order 2. When we want to talk about probabilities, say P( X n ? B) ? P( X ? B), we need to exploit results based on convergence in distribution. Examples 14.8 and 14.9 are important applications that require both convergence in mean of order 2 and convergence in distribution. We must also mention that the central limit theorem, which we made extensive use of in Chapter 6 on confidence intervals, is a statement about convergence in distribution. Convergence in probability is a concept we have also been using for quite a while, e.g., the weak law of large numbers in Section 3.3.
The present chapter is devoted to the study of convergence in mean of order p, while the remaining types of...
