TITLE:
Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions
AUTHORS:
Frederic von Wegner
KEYWORDS:
Heavy-Tailed Distributions, Random Sampling, Gaussian, Exponential, Power-Law
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.13,
July
18,
2014
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.