Power-Law Distributions: Beyond Paretian Fractality

TL;DR

This paper broadens the understanding of fractal probability distributions by identifying six classes with power-law structures, unified through Poisson processes with various intrinsic scales, extending beyond the traditional Paretian view.

Contribution

It introduces a comprehensive framework showing six classes of fractal distributions, all with power-law structures, as fixed points of renormalizations, linked to underlying Poisson processes.

Findings

Identifies six classes of fractal probability laws with power-law structures.

Shows these classes are fixed points of renormalization transformations.

Provides a unified Poisson process-based theory for fractal distributions.

Abstract

The notion of fractality, in the context of positive-valued probability distributions, is conventionally associated with the class of Paretian probability laws. In this research we show that the Paretian class is merely one out of six classes of probability laws - all equally entitled to be ordained fractal, all possessing a characteristic power-law structure, and all being the unique fixed points of renormalizations acting on the space of positive-valued probability distributions. These six fractal classes are further shown to be one-dimensional functional projections of underlying fractal Poisson processes governed by: (i) a common elemental power-law structure; and, (ii) an intrinsic scale which can be either linear, harmonic, log-linear, or log-harmonic. This research provides a panoramic and comprehensive view of fractal distributions, backed by a unified theory of their underlying Poissonian fractals.