6 - Characteristic function

Abstract

This chapter describes the general properties of a characteristic function. An important tool in the study of r.v.'s and their p.m.'s or d.f.'s is the characteristic function (ch.f.). In analysis, the ch.f. is known as the Fourier–Stieltjes transform of μ or F. It can also be defined over a wider class of μ or F and, furthermore, be considered as a function of a complex variable t, under certain conditions that ensure the existence of the integrals. This extension is important in some application. The binary operation of convolution * is commutative and associative. For the corresponding binary operation of addition of independent r.v.'s has these two properties. The convolution of two probability density functions p1 and p2 is defined to be the probability density function. The convolution of two absolutely continuous d.f.'s with densities p1 and p2 is absolutely continuous with density p1* p2.