{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T18:59:43Z","timestamp":1781204383664,"version":"3.54.1"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T00:00:00Z","timestamp":1735776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801307"],"award-info":[{"award-number":["11801307"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2017MA012"],"award-info":[{"award-number":["ZR2017MA012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2021MA009"],"award-info":[{"award-number":["ZR2021MA009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["LWCXS202223"],"award-info":[{"award-number":["LWCXS202223"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["11801307"],"award-info":[{"award-number":["11801307"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["ZR2017MA012"],"award-info":[{"award-number":["ZR2017MA012"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["ZR2021MA009"],"award-info":[{"award-number":["ZR2021MA009"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["LWCXS202223"],"award-info":[{"award-number":["LWCXS202223"]}]},{"name":"Postgraduate Dissertation Research Innovation Foundation of Qufu Normal University","award":["11801307"],"award-info":[{"award-number":["11801307"]}]},{"name":"Postgraduate Dissertation Research Innovation Foundation of Qufu Normal University","award":["ZR2017MA012"],"award-info":[{"award-number":["ZR2017MA012"]}]},{"name":"Postgraduate Dissertation Research Innovation Foundation of Qufu Normal University","award":["ZR2021MA009"],"award-info":[{"award-number":["ZR2021MA009"]}]},{"name":"Postgraduate Dissertation Research Innovation Foundation of Qufu Normal University","award":["LWCXS202223"],"award-info":[{"award-number":["LWCXS202223"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper introduces U-S chance spaces, a new framework based on uncertainty theory and sub-linear expectation theory, to depict human uncertainty and sub-linear features, simultaneously. These spaces can be used to analyze the characteristics of uncertain random variables and study investments and other related issues in incomplete financial markets. Within the framework, sub-linear expectation theory describes the randomness in financial behaviors, while uncertainty theory describes the uncertainty associated with government macro-control or experts\u2019 opinions. The main achievement of this paper is the derivation of the Kolmogorov law of large numbers for uncertain random variables under U-S chance spaces. Examples are provided, and the theorems can be applied to uncertain random variables that are functions of random variables with symmetric or asymmetric distributions and uncertain variables with symmetric or asymmetric distributions. In some cases, when both random and uncertain variables are symmetric, the limit in the law exhibits the form that is characterized by symmetrical uncertain variables.<\/jats:p>","DOI":"10.3390\/sym17010062","type":"journal-article","created":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T10:32:26Z","timestamp":1735813946000},"page":"62","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Laws of Large Numbers for Uncertain Random Variables in the Framework of U-S Chance Theory"],"prefix":"10.3390","volume":"17","author":[{"given":"Xiaoting","family":"Fu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Weifang University, Weifang 261061, China"},{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xue","family":"Meng","sequence":"additional","affiliation":[{"name":"School of Management, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Economics, Ocean University of China, Qingdao 266100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Deguo","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"131","DOI":"10.5802\/aif.53","article-title":"Theory of capacities","volume":"5","author":"Choquet","year":"1954","journal-title":"Ann. 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