Chapter Eight

Characteristic Function

# 8.1 Introduction/Purpose of the Chapter

The characteristic function of a random variable is a powerful tool for analyzing the distribution of sums of independent random variables. To some readers, characteristic functions may already be familiar in a different form: If a random variable is continuous and thus it has a probability density function f(x), then its characteristic function is the Fourier transform of the function f(x).

# 8.2 Vignette/Historical Notes

According to Kenney (1942) and (Todhunter, 1865, pp. 309–313), the first use of an analytic method substantially equivalent to the characteristic functions is due to Joseph Louis de Lagrange in his work Réflexions sur la résolution algébrique des équations, published around 1770 de Lagrange (1770). In this work, de Lagrange introduces a simple form of the Fourier transform. The Fourier transform and the Fourier series was properly introduced and refined by Joseph Fourier in his Mémoire sur la propagation de la chaleur dans les corps solides. He applies the series to find the solution for the heat equation.

The first general definition of the characteristic function is due to Pierre-Simon marquis de Laplace in his classic “Théorie analytique des probabilités” (Laplace, 1812, pp. 83–84). He first introduces the generating function as “fonction generatrice” (described in the following chapter on moment generating function). Laplace is recognized with the introduction of the moment-generating ...