Fourier Theory of Probability Distributions

Abstract

Characteristic functions, that is Fourier transforms of probability measures, play a major role in Probability theory, in particular in the Fourier theory of wide-sense stationary stochastic processes, whose starting point is the notion of power spectral measure. It turns out that the existence of such a measure is a direct consequence of Bochner’s theorem of characterization of characteristic functions, and that the proof of its unicity is a rephrasing of Paul Lévy’s inversion theorem. Another result of Paul Lévy, characterizing convergence in distribution in terms of characteristic functions, intervenes in an essential way in the proof of Bochner’s theorem. In fact, characteristic functions are the link between the Fourier theory of deterministic functions and that of stochastic processes. This chapter could have been entitled “Convergence in distribution of random sequences”, a classical topic of probability theory. However, we shall need to go slightly beyond this and give the extension of Paul Lévy’s convergence theorem to sequences of finite measures (instead of probability distributions) as this is needed in Chap. 5 for the proof of existence of the Bartlett spectral measure.