Eliazar Iddo, Klafter Joseph
Department of Technology Management, Holon Institute of Technology, PO Box 305, Holon 58102, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Jun;77(6 Pt 1):061125. doi: 10.1103/PhysRevE.77.061125. Epub 2008 Jun 18.
We explore six classes of fractal probability laws defined on the positive half-line: Weibull, Frechét, Lévy, hyper Pareto, hyper beta, and hyper shot noise. Each of these classes admits a unique statistical power-law structure, and is uniquely associated with a certain operation of renormalization. All six classes turn out to be one-dimensional projections of underlying Poisson processes which, in turn, are the unique fixed points of Poissonian renormalizations. The first three classes correspond to linear Poissonian renormalizations and are intimately related to extreme value theory (Weibull, Frechét) and to the central limit theorem (Lévy). The other three classes correspond to nonlinear Poissonian renormalizations. Pareto's law--commonly perceived as the "universal fractal probability distribution"--is merely a special case of the hyper Pareto class.
威布尔定律、弗雷歇定律、列维定律、超帕累托定律、超贝塔定律和超散粒噪声定律。这些类别中的每一个都具有独特的统计幂律结构，并且与某种重整化操作唯一相关。事实证明，所有这六类都是基础泊松过程的一维投影，而泊松过程又是泊松重整化的唯一不动点。前三类对应于线性泊松重整化，并且与极值理论（威布尔定律、弗雷歇定律）和中心极限定理（列维定律）密切相关。其他三类对应于非线性泊松重整化。帕累托定律——通常被视为“通用分形概率分布”——仅仅是超帕累托类别的一个特例。
