Characteristic Functions

Abstract

Adding independent random variables is a frequent occupation in probability theory. Mathematically this corresponds to convolving functions. Just as there are Fourier transforms and Laplace transforms which transform convolution into multiplication, there are transforms in probability theory that transform addition of independent random variables into multiplication of transforms. Although we shall mainly use one of them, the characteristic function, we shall, in this chapter, briefly also present three others — the cumulant generating function, which is the logarithm of the characteristic function; the probability generating function; and the moment generating function. In Chapter 5 we shall prove so-called continuity theorems, which permit limits of distributions to be determined with the aid of limits of transforms.