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The retained closure property simplifies marginal properties such as moments. The family can be obtained by conditioning independent infinitely divisible distributions on the total and also by mixing the multinomial distribution with the normalized infinitely divisible distribution. The closure property justifies a stochastic process of the family by Kolmogorov\u2019s extension theorem. Over time, Gibbs partitions of a positive integer appear as the limiting distributions of the family.<\/jats:p>","DOI":"10.1017\/jpr.2025.10042","type":"journal-article","created":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T10:22:33Z","timestamp":1763979753000},"page":"501-535","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Infinitely resizable urn models"],"prefix":"10.1017","volume":"63","author":[{"given":"Nobuaki","family":"Hoshino","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hwp6a56","id-type":"ROR","asserted-by":"publisher"}],"name":"Kanazawa University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,11,24]]},"reference":[{"key":"S0021900225100429_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(93)90027-G"},{"key":"S0021900225100429_ref37","doi-asserted-by":"publisher","DOI":"10.1198\/016214505000000132"},{"key":"S0021900225100429_ref12","volume-title":"Lagrangian Probability Distributions","author":"Consul","year":"2006"},{"key":"S0021900225100429_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3117-7"},{"key":"S0021900225100429_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8643-8"},{"key":"S0021900225100429_ref40","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/80.2.363"},{"key":"S0021900225100429_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2011.03.015"},{"key":"S0021900225100429_ref51","first-page":"893","article-title":"D","volume":"1","author":"Steutel","year":"1979","journal-title":"Ann. 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