October 2014
Intermediate to advanced
576 pages
14h 32m
English
Content preview from Probability and Stochastic Processes
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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 ...
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