Sums of Independent Random Variables

Chapters 9 and 10 are much less elementary in character than the first eight chapters of this book. They constitute an introduction to the limit theorems of probability theory and to the role of characteristic functions in probability theory. These chapters seek to provide a careful and rigorous derivation of the law of large numbers and the central limit theorem.

In this chapter we treat the problem of finding the probability law of a random variable that arises as the sum of independent random variables. A major tool in this study is the characteristic function of a random variable, introduced in section 2. In section 3 it is shown that the probability law of a random variable can be determined from its characteristic function. Section 4 discusses some consequences of the basic result that the characteristic function of a sum of independent random variables is the product of the characteristic functions of the individual random variables. Section 5 gives the proofs of the inversion formulas stated in section 3.