{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T08:12:34Z","timestamp":1780474354372,"version":"3.54.1"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2017,8,17]],"date-time":"2017-08-17T00:00:00Z","timestamp":1502928000000},"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>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.<\/jats:p>","DOI":"10.3390\/e19080417","type":"journal-article","created":{"date-parts":[[2017,8,21]],"date-time":"2017-08-21T04:12:17Z","timestamp":1503288737000},"page":"417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes"],"prefix":"10.3390","volume":"19","author":[{"given":"Arthur","family":"Sousa","sequence":"first","affiliation":[{"name":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]},{"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, Z\u00fcrich 8092, Switzerland"},{"name":"Swiss Finance Institute, University of Geneva, Geneva 1211, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Misako","family":"Takayasu","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1137\/070710111","article-title":"Power-law distributions in empirical data","volume":"51","author":"Clauset","year":"2009","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mitchell, M. 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