Affiliations: Department of Technology Management, Holon Institute of Technology, Holon, Israel | School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel | Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, Freiburg, Germany. E-mails: {eliazar, klafter}@post.tau.ac.il
Note: [] Corresponding author: Iddo Eliazar, Department of Technology Management, Holon Institute of Technology, P.O. Box 305, Holon 58102, Israel. E-mail: eliazar@post.tau.ac.il.
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.
Keywords: Paretian fractality, renormalization, Poisson processes, Poissonian fractality and renormalization, Fréchet, Weibull and Lévy stable distributions
