Characteristic Functions of Functions

Abstract

BECAUSE the expense of computing Fourier transforms is much less than it used to be, the definition of statistical distributions by means of their characteristic functions (CFs) is of increased practical value. I shall outline a formal process for expressing the characteristic function of a function g(X) in terms of the CF of X, denoted by φ, together with some examples and deductions. A further formalism relates the Mellin transforms of the CF and the PD (probability density) of X. When applied to multivariate stable distributions the formalisms lead to generalizations of parts of multivariate statistical analysis most of which has been classically based on the multigaussian distribution.