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Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions

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DOI: 10.4236/am.2014.513198    2,865 Downloads   3,666 Views  

ABSTRACT

A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. The model considers radially symmetric Gaussian, exponential and power-law functions inn= 1, 2, 3 dimensions. Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. For two-dimensional Gaussians and one-dimensional exponential functions, exact power-laws with exponent 1 are obtained. In other cases, densities with an approximate power-law behaviour close to the origin arise. These densities are analyzed using Padé approximants in order to show the approximate power-law behaviour. If the sampled function itself follows a power-law with exponent α, random sampling leads to densities that also follow an exact power-law, with exponent -n/a 1. The presented mechanism shows that power-laws can arise in generic situations different from previously considered specialized systems such as multi-particle systems close to phase transitions, dynamical systems at bifurcation points or systems displaying self-organized criticality. Thus, the presented mechanism may serve as an alternative hypothesis in system identification problems.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Wegner, F. (2014) Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions. Applied Mathematics, 5, 2050-2056. doi: 10.4236/am.2014.513198.

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