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Open AccessArticle

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

1
Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan
2
Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan
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Sony Computer Science Laboratories, Tokyo 141-0022, Japan
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Department of Management, Technology and Economics, ETH Zürich, Zürich 8092, Switzerland
5
Swiss Finance Institute, University of Geneva, Geneva 1211, Switzerland
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(8), 417; https://doi.org/10.3390/e19080417
Received: 2 August 2017 / Revised: 13 August 2017 / Accepted: 16 August 2017 / Published: 17 August 2017
(This article belongs to the Special Issue Statistical Mechanics of Complex and Disordered Systems)
We introduce a simple growth model in which the sizes of entities evolve as multiplicative random processes that start at different times. A novel aspect we examine is the dependence among entities. For this, we consider three classes of dependence between growth factors governing the evolution of sizes: independence, Kesten dependence and mixed dependence. We take the sum X of the sizes of the entities as the representative quantity of the system, which has the structure of a sum of product terms (Sigma-Pi), whose asymptotic distribution function has a power-law tail behavior. We present evidence that the dependence type does not alter the asymptotic power-law tail behavior, nor the value of the tail exponent. However, the structure of the large values of the sum X is found to vary with the dependence between the growth factors (and thus the entities). In particular, for the independence case, we find that the large values of X are contributed by a single maximum size entity: the asymptotic power-law tail is the result of such single contribution to the sum, with this maximum contributing entity changing stochastically with time and with realizations. View Full-Text
Keywords: power-law; random multiplicative process; stochastic process; growth model; dependence power-law; random multiplicative process; stochastic process; growth model; dependence
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Sousa, A.M.Y.R.; Takayasu, H.; Sornette, D.; Takayasu, M. Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes. Entropy 2017, 19, 417.

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