Escort mean values and the characterization of power-law-decaying probability densities

TL;DR

This paper explores how escort mean values can characterize probability densities with power-law decay, especially when standard moments diverge, and discusses related mathematical tools like the q-Fourier Transform.

Contribution

It extends the characterization of probability densities using escort mean values to cases with divergent standard moments, relevant for power-law distributions.

Findings

Escort mean values can characterize power-law decaying densities.

Standard moments diverge for certain distributions like the Cauchy-Lorentz.

Discusses mathematical aspects of the q-Fourier Transform.

Abstract

Escort mean values (or $q$-moments) constitute useful theoretical tools for describing basic features of some probability densities such as those which asymptotically decay like {\it power laws}. They naturally appear in the study of many complex dynamical systems, particularly those obeying nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. They recover standard mean values (or moments) for $q=1$. Here we discuss the characterization of a (non-negative) probability density by a suitable set of all its escort mean values together with the set of all associated normalizing quantities, provided that all of them converge. This opens the door to a natural extension of the well known characterization, for the $q=1$ instance, of a distribution in terms of the standard moments, provided that {\it all} of them have {\it finite} values. This question would be specially relevant in connection with probability densities having {\it divergent} values for all nonvanishing standard moments higher than a given one (e.g., probability densities asymptotically decaying as power-laws), for which the standard approach is not applicable. The Cauchy-Lorentz distribution, whose second and higher even order moments diverge, constitutes a simple illustration of the interest of this investigation. In this context, we also address some mathematical subtleties with the aim of clarifying some aspects of an interesting non-linear generalization of the Fourier Transform, namely, the so-called $q$-Fourier Transform.