Generating functions were introduced by the Swiss genius Leonhard Euler (1707–1783) in the eighteenth century to facilitate calculations in counting problems. However, this important concept is also extremely useful in applied probability, as was first demonstrated by the work of Abraham de Moivre (1667–1754) who discovered the technique of generating functions independently of Euler. In modern probability theory, generating functions are an indispensable tool in combination with methods from numerical analysis.

The purpose of this chapter is to give the basic properties of generating functions and to show the utility of this concept. First, the generating function is defined for a discrete random variable on nonnegative integers. Next, we consider the more general moment-generating function, which is defined for any random variable. The (moment) generating function is a powerful tool for both theoretical and computational purposes. In particular, it can be used to prove the central limit theorem. A sketch of the proof will be given. This chapter also gives a proof of the strong law of large numbers, using moment-generating functions together with so-called Chernoff bounds. Finally, the strong law of large numbers is used to establish the powerful renewal-reward theorem for stochastic processes having the property that the process probabilistically restarts itself at certain points in time.

Generating functions

We first introduce the concept of generating function for a discrete random variable X whose possible values belong to the set of nonnegative integers.