Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes
De Sousa, Arthur M.Y.R.
- Journal Article
Rights / licenseCreative Commons Attribution 4.0 International
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. Show more
Journal / seriesEntropy
Pages / Article No.
SubjectPower-law; random multiplicative process; Stochastic process; Growth model; Dependence
Organisational unit03738 - Sornette, Didier / Sornette, Didier
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