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

0545603 - ÚI 2022 CH eng J - Journal Article
Yamashita Rios de Sousa, Arthur Matsuo - Takayasu, H. - Sornette, D. - Takayasu, M.
Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes.
Entropy. Roč. 19, č. 8 (2017), č. článku 417. E-ISSN 1099-4300
Keywords : growth * fluctuations * explanation * dynamics * internet * power-law * random multiplicative process * stochastic process * growth model * dependence
Impact factor: 2.305, year: 2017

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.
Permanent Link: http://hdl.handle.net/11104/0322282