CHAPTER 6 - The Central Limit Theorem

Abstract

This chapter discusses the central limit theorem. The central limit theorem refers to a body of theorems, each of which deals with conditions under which distribution functions of sums of random variables converge. The limit distributions have a special property, called infinite divisibility. The chapter discusses the infinitely divisible distribution functions. A distribution function F is said to be infinitely divisible, if for every positive integer n, there is a distribution function Fn such that F is the n-fold convolution of Fn with itself. There are the two important examples of infinitely divisible distribution functions: (1) the normal distribution and (2) the Poisson distribution. If F is an infinitely divisible distribution function, then its characteristic function f never vanishes. The chapter presents a derivation of the Lévy–Khinchine representation of the characteristic function of an infinitely divisible distribution function.