Fourier and Cauchy-Stieltjes transforms of power laws including stable distributions

TL;DR

This paper introduces a new class of probability measures characterized by their Fourier and Cauchy-Stieltjes transforms, encompassing stable distributions and extending to non-commutative probability theories.

Contribution

It defines a novel class of distributions with power series Fourier transforms including real powers, unifying stable distributions in probability and non-commutative probability.

Findings

Class includes distributions with Pareto-like tails.

Fourier transform characterized by real power series.

Stable distributions are included or excluded based on the setting.

Abstract

We introduce a class of probability measures whose densities near infinity are mixtures of Pareto distributions. This class can be characterized by the Fourier transform which has a power series expansion including real powers, not only integer powers. This class includes stable distributions in probability and also non-commutative probability theories. We also characterize the class in terms of the Cauchy-Stieltjes transform and the Voiculescu transform. If the stability index is greater than one, stable distributions in probability theory do not belong to that class, while they do in non-commutative probability.