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Fractal probability laws.

Iddo Eliazar¹, Joseph Klafter

¹Department of Technology Management, Holon Institute of Technology, PO Box 305, Holon 58102, Israel. eliazar@post.tau.ac.il

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Summary

This summary is machine-generated.

This study reveals six fractal probability laws are projections of Poisson processes, linked by unique renormalization operations. These laws connect to extreme value theory and central limit theorem, with Pareto

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Area of Science:

Probability Theory
Statistical Mechanics
Fractal Geometry

Background:

Fractal probability laws exhibit power-law structures.
Renormalization operations are key to understanding these distributions.
Existing research often focuses on specific laws like Pareto's.

Purpose of the Study:

To explore six classes of fractal probability laws on the positive half-line.
To identify the underlying structure and renormalization operations for each class.
To connect these laws to fundamental concepts like extreme value theory and the central limit theorem.

Main Methods:

Analysis of six fractal probability law classes: Weibull, Frechét, Lévy, hyper Pareto, hyper beta, and hyper shot noise.
Investigation of their unique statistical power-law structures.
Examination of their relationship to underlying Poisson processes and Poissonian renormalizations.

Main Results:

All six classes are one-dimensional projections of underlying Poisson processes.
Each class is uniquely associated with a specific renormalization operation.
The first three classes (Weibull, Frechét, Lévy) relate to linear Poissonian renormalizations, extreme value theory, and the central limit theorem.
The other three classes involve nonlinear Poissonian renormalizations.
Pareto's law is identified as a special case of the hyper Pareto class.

Conclusions:

Fractal probability laws are unified under Poisson processes and renormalization.
This framework provides a deeper understanding of distributions like Weibull, Frechét, Lévy, and Pareto.
The study highlights the significance of linear and nonlinear Poissonian renormalizations.