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


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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Hohenberg, P.C. and Halperin, B.I. (1977) Theory of Dynamic Critical Phenomena. Reviews of Modern Physics, 49, 435-479.
[2] Mitzenmacher, M. (2003) A Brief History of Generative Models for Power Law and Lognormal Distributions. Internet Mathematics, 1, 226-251.
[3] Montroll, M. and Shlesinger, M.F. (1983) Maximum Entropy Formalism, Fractals, Scaling Phenomena, and 1/f Noise: A Tale of Tails. The Journal of Chemical Physics, 32, 209-230.
[4] Newman, M.E.J. (2005) Power Laws, Pareto Distributions and Zipfs Law, Contemporary Physics.
[5] Stanley, H.E. (1999) Scaling, Universality, and Renormalization: Three Pillars of Modern Critical Phenomena. Reviews of Modern Physics, 71, S358-S366.
[6] Sornette, D. (2004) Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization, and Disorder: Concepts and Tools. Springer, New York.
[7] Malmgren, R.D., Stouffer, D.B., Motter, A.E. and Amaral, L.A.N. (2008) A Poissonian Explanation for Heavy Tails in E-Mail Communication. Proceedings of National Academy Science of USA, 105, 18153-18158.
[8] Sornette, D. (1998) Multiplicative Processes and Power Laws. Physical Review E, 57, 4811-4813.
[9] Touboul, J. and Destexhe, A. (2010) Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics? PLoS One, 5, e8982.
[10] Ramachandran, K.M. and Tsokos, C.P. (2009) Mathematical Statistics with Applications. Academic Press.
[11] Jr. Baker, G.A. and Graves-Morris, P. (1996) Padé Approximants. Cambridge University Press, New York.
[12] Ríos, E., Shirokova, N., Kirsch, W.G., Pizarro, G., Stern, M.D., Cheng, H. and González, A. (2001) A Preferred Amplitude of Calcium Sparks in Skeletal Muscle. Biophysical Journal, 80, 169-183.

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