TY - JOUR
T1 - Power-law distributions
T2 - Beyond Paretian fractality
AU - Eliazar, Iddo
AU - Klafter, Joseph
PY - 2009
Y1 - 2009
N2 - 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.
AB - 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.
KW - Fréchet
KW - Paretian fractality
KW - Poisson processes
KW - Poissonian fractality and renormalization
KW - Weibull and Lévy stable distributions
KW - renormalization
UR - http://www.scopus.com/inward/record.url?scp=77954529514&partnerID=8YFLogxK
U2 - 10.3233/RDA-2009-0014
DO - 10.3233/RDA-2009-0014
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AN - SCOPUS:77954529514
SN - 1569-7371
VL - 1
SP - 155
EP - 170
JO - Risk and Decision Analysis
JF - Risk and Decision Analysis
IS - 3
ER -