Probability and Stochastic Processes

Chapter 7Limit Theorems

Introduction

In the previous chapter, we talked about a sequence of distribution functions converging to a target distribution, which is, as we shall see, the concept of convergence in distribution. It turns out that this type of convergence is one of the most useful concepts in applications. Often, we hypothesize a model which will have parameters estimated using statistics obtained from real data. It is essential that the approximations (statistics) converge to the desired parameters, and it is crucial that they do so fast. The way in which they converge and the rate of convergence are covered in this chapter. The chapter is dedicated to the review and discovery of the most important types of convergence in probability theory.

7.1 Types of Convergence

Let be a probability space, and let be a sequence of random variables. Furthermore, let be a target random variable. Throughout this chapter, we shall take to denote the index of the sequence of random variables. In general, the sequence may be indexed by a different index set, the only condition being ...