On the fractal characterization of Paretian Poisson processes |
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Authors: | Iddo I Eliazar Igor M Sokolov |
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Institution: | 1. Department of Technology Management, Holon Institute of Technology, P.O.B. 305, Holon 58102, Israel;2. Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany |
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Abstract: | Paretian Poisson processes are Poisson processes which are defined on the positive half-line, have maximal points, and are quantified by power-law intensities. Paretian Poisson processes are elemental in statistical physics, and are the bedrock of a host of power-law statistics ranging from Pareto’s law to anomalous diffusion. In this paper we establish evenness-based fractal characterizations of Paretian Poisson processes. Considering an array of socioeconomic evenness-based measures of statistical heterogeneity, we show that: amongst the realm of Poisson processes which are defined on the positive half-line, and have maximal points, Paretian Poisson processes are the unique class of ‘fractal processes’ exhibiting scale-invariance. The results established in this paper are diametric to previous results asserting that the scale-invariance of Poisson processes–with respect to physical randomness-based measures of statistical heterogeneity–is characterized by exponential Poissonian intensities. |
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Keywords: | Gini&rsquo s index Pietra&rsquo s index Evenness ratio Min&ndash max ratio Moment ratio Power-laws |
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