Characteristic Functions

Abstract

The characteristic function of a random variable is a complex-valued function calculated from its distribution function, but is more tractable in many ways, primarily because of its superior smoothness properties. The characteristic function uniquely determines the distribution function, so that recognizing the characteristic function of a random variable identifies its distribution function. The density function, if it exists, can be recovered algorithmically from the characteristic function. Characteristic functions convert convolution to the simpler operation of pointwise multiplication. Moments of a random variable are derivatives at zero of its characteristic function, while existence of even-order derivatives of the characteristic function implies existence of the corresponding moments. Finally, random variables converge in distribution if and only if their characteristic functions converge pointwise.