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 {\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.

20 pages (2 Appendices have been added)

References in corpus (11)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• Defect turbulence and generalized statistical mechanics
• Asymptotics of superstatistics
• Statistics of 3-dimensional Lagrangian turbulence
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A General Nonlinear Fokker-Planck Equation and its Associated Entropy
• Nonergodicity and Central Limit Behavior for Long-range Hamiltonians
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product
• On a representation of the inverse Fq transform
• Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

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• On the average uncertainty for systems with nonlinear coupling
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