{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T16:17:26Z","timestamp":1773159446894,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,19]],"date-time":"2021-02-19T00:00:00Z","timestamp":1613692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.<\/jats:p>","DOI":"10.3390\/e23020241","type":"journal-article","created":{"date-parts":[[2021,2,19]],"date-time":"2021-02-19T20:54:08Z","timestamp":1613768048000},"page":"241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Sigma-Pi Structure with Bernoulli Random Variables: Power-Law Bounds for Probability Distributions and Growth Models with Interdependent Entities"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3813-1295","authenticated-orcid":false,"given":"Arthur Matsuo","family":"Yamashita Rios de Sousa","sequence":"first","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4452-5517","authenticated-orcid":false,"given":"Hideki","family":"Takayasu","sequence":"additional","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Sony Computer Science Laboratories, Tokyo 141-0022, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7874-7962","authenticated-orcid":false,"given":"Didier","family":"Sornette","sequence":"additional","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Department of Management, Technology and Economics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland"},{"name":"Swiss Finance Institute, University of Geneva, 1211 Geneva, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2467-614X","authenticated-orcid":false,"given":"Misako","family":"Takayasu","sequence":"additional","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sousa, A.M.Y.R., Takayasu, H., Sornette, D., and Takayasu, M. 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