Characteristic Functions and Applications

Abstract

Characteristic functions were first systematically studied by Paul Lévy, although they were used by others before him. It provides an extremely powerful tool in probability in general, and in asymptotic theory in particular. The power of the characteristic function derives from a set of highly convenient properties. Like the mgf, it determines a distribution. But unlike mgfs, existence is not an issue, and it is a bounded function. It is easily transportable for common functions of random variables, such as convolutions. And it can be used to prove convergence of distributions, as well as to recognize the name of a limiting distribution. It is also an extremely handy tool in proving characterizing properties of distributions. For instance, the Cramér–Levy theorem (see Chapter 1), which characterizes a normal distribution, has so far been proved by only using characteristic function methods. There are two disadvantages in working with characteristic functions. First, it is a complex-valued function, in general, and so, familiarity with basic complex analysis is required. Second, characteristic function proofs usually do not lead to any intuition as to why a particular result should be true. All things considered, knowledge of basic characteristic function theory is essential for statisticians, and certainly for students of probability.