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

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 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 nonnegative 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 all of them have finite values. This question would be specially relevant in connection with probability densities having 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 nonlinear generalization of the Fourier transform, namely, the so called q-Fourier transform.