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From the central limit theorem to heavy-tailed distributions

Part of: Stochastic processes Distribution theory - Probability Distribution theory Limit theorems

Published online by Cambridge University Press:  14 July 2016

Jinwen Chen
Jinwen Chen*
Affiliation:
Tsinghua University
*
∗Postal address: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. Email address: jchen@math.tsinghua.edu.cn
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Abstract

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

Keywords

Heavy-tailed distributionpower lawcentral limit theoremrandomly indexed product

MSC classification

Primary: 60E05: Distributions 60F15: Strong theorems 60F99: None of the above, but in this section 60G50: Sums of independent random variables; random walks
Secondary: 62E10: Characterization and structure theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 803 - 806
Copyright
Copyright © Applied Probability Trust 2003 

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References

Gong, W. et al. (2001). On the tails of web file size distributions. Preprint, Department of Electrical and Computer Engineering, University of Massachusetts, Amherst.Google Scholar
Korolev, V. Yu. (1994). Convergence of random sequences with independent random indices. I. Theory Prob. Appl. 39, 282–297.CrossRefGoogle Scholar
Mitzenmacher, M. (2001). A brief history of generative models for power law and lognormal distributions. In Proc. 39th Allerton Conf. Commun. Control Comput. (Monticello, IL, October 2001).Google Scholar
Montroll, E. W., and Shlesinger, M. F. (1982). On 1/f noise and other distributions with long tails. Proc. Nat. Acad. Sci. USA 79, 3380–3383.CrossRefGoogle ScholarPubMed
Montroll, E. W., and Shlesinger, M. F. (1983). Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J. Statist. Phys. 32, 209–230.CrossRefGoogle Scholar
Reed, W. J. (2001). The double Pareto-lognormal distribution–a new parametric model for size distribution. Preprint, Department of Mathematics and Statistics, University of Victoria.Google Scholar
Reed, W. J. (2003). The Pareto law of incomes–an explanation and an extension. Physica A 319, 469–486.CrossRefGoogle Scholar