Characteristic Functions

TL;DR: Characteristic functions are used to study properties of random variables and their distributions. They are defined and studied in this chapter, with applications to conditional distributions and infinite divisibility.

Abstract: This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.