{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:34:49Z","timestamp":1760236489069,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T00:00:00Z","timestamp":1637971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz\u2013Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz\u2013Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.<\/jats:p>","DOI":"10.3390\/sym13122258","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:40Z","timestamp":1638328360000},"page":"2258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Some Laws of Large Numbers for Uncertain Random Variables"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9325-4493","authenticated-orcid":false,"given":"Piotr","family":"Nowak","sequence":"first","affiliation":[{"name":"Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9877-508X","authenticated-orcid":false,"given":"Olgierd","family":"Hryniewicz","sequence":"additional","affiliation":[{"name":"Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Rie\u010dan, B., and Neubrunn, T. (1997). Integral, Measure, and Ordering, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8919-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (2012). On Intuitionistic Fuzzy Sets Theory, Springer.","DOI":"10.1007\/978-3-642-29127-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1002\/mana.19750680109","article-title":"Intervall-wertige Mengen","volume":"68","author":"Jahn","year":"1975","journal-title":"Math. Nachrichten"},{"key":"ref_5","first-page":"1","article-title":"Introduction to Grey system theory","volume":"1","author":"Deng","year":"1989","journal-title":"J. Grey Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-0255(78)90019-1","article-title":"Fuzzy random variables\u2014I. definitions and theorems","volume":"15","author":"Kwakernaak","year":"1978","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0022-247X(86)90093-4","article-title":"Fuzzy random variables","volume":"114","author":"Puri","year":"1986","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Couso, I., Dubois, D., and S\u00e1nchez, L. (2014). Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables, Springer.","DOI":"10.1007\/978-3-319-08611-8"},{"key":"ref_9","first-page":"765","article-title":"Generalized versions of MV-algebraic central limit theorems","volume":"51","author":"Nowak","year":"2018","journal-title":"Kybernetika"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.ins.2016.03.052","article-title":"On generalized versions of central limit theorems for IF-events","volume":"355\u2013356","author":"Nowak","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2471","DOI":"10.1007\/s00500-017-2731-3","article-title":"On central limit theorems for IV-events","volume":"22","author":"Nowak","year":"2018","journal-title":"Soft Comput."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"\u010cunderl\u00edkov\u00e1, K. (2020). Martingale convergence theorem for the conditional intuitionistic fuzzy probability. Mathematics, 8.","DOI":"10.3390\/math8101707"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s00500-012-0935-0","article-title":"Uncertain random variables: A mixture of uncertainty and randomness","volume":"17","author":"Liu","year":"2013","journal-title":"Soft Comput."},{"key":"ref_14","unstructured":"Liu, B. (2007). Uncertainty Theory, Springer. [2nd ed.]."},{"key":"ref_15","first-page":"3","article-title":"Some research problems in uncertainty theory","volume":"3","author":"Liu","year":"2009","journal-title":"J. Uncertain Syst."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chow, Y.S., and Teicher, H. (1997). Probability Theory Independence, Interchangeability, Martingales, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4612-1950-7"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/0165-0114(84)90033-2","article-title":"A strong law of large numbers for fuzzy random variables","volume":"12","author":"Miyakoshi","year":"1984","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1098\/rspa.1986.0091","article-title":"Limit theorems for fuzzy random variables","volume":"407","author":"Klement","year":"1986","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/0165-0114(94)90161-9","article-title":"Marcinkiewicz-type law of large numbers for fuzzy random variables","volume":"64","author":"Hong","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.spl.2005.04.030","article-title":"On Chung\u2019s type law of large numbers for fuzzy random variables","volume":"74","author":"Joo","year":"2005","journal-title":"Stat. Probab. Lett."},{"key":"ref_21","first-page":"307","article-title":"Maximal inequalities and some convergence theorems for fuzzy random variables","volume":"52","author":"Ahmadzade","year":"2016","journal-title":"Kybernetika"},{"key":"ref_22","unstructured":"Pap, E. (2002). Probability on MV-algebras. Handbook of Measure Theory, Elsevier."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Nowak, P., and Hryniewicz, O. (2019). On MV-algebraic versions of the strong law of large numbers. Entropy, 21.","DOI":"10.3390\/e21070710"},{"key":"ref_24","unstructured":"De Baets, B., De Caluwe, R., De Tre, G., Fodor, J., Kacprzyk, J., and Zadro\u017cny, S. (2004). Weak law of large numbers for IF-events. Current Issues in Data and Knowledge Engineering, Exit."},{"key":"ref_25","first-page":"30","article-title":"Laws of large number for M-observables","volume":"13","year":"2007","journal-title":"Notes IFS"},{"key":"ref_26","unstructured":"Atanassov, K.T., Atanassova, V., Kacprzyk, J., Ka\u0142uszko, A., Krawczak, M., Owsi\u0144ski, J.W., Sotirov, S., Sotirova, E., Szmidt, E., and Zadro\u017cny, S. (2021). Convergence of intuitionistic fuzzy observables. Uncertainty and Imprecision in Decision Making and Decision Support: New Challenges, Solutions and Perspectives, Springer International Publishing."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2293","DOI":"10.1109\/TFUZZ.2019.2896849","article-title":"Strong laws of large numbers for IVM-events","volume":"27","author":"Nowak","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1109\/TFUZZ.2015.2466080","article-title":"Law of large numbers for uncertain random variables","volume":"24","author":"Yao","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.3233\/IFS-162187","article-title":"Law of large numbers for uncertain random variables with different chance distributions","volume":"31","author":"Gao","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1109\/TFUZZ.2017.2724021","article-title":"Convergence in distribution for uncertain random variables","volume":"26","author":"Gao","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5655","DOI":"10.1007\/s00500-017-2586-7","article-title":"A stronger law of large numbers for uncertain random variables","volume":"22","author":"Sheng","year":"2018","journal-title":"Soft Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/s11424-020-9192-y","article-title":"Convergence in distribution for uncertain random sequences with dependent random variables","volume":"34","author":"Gao","year":"2021","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/2195-5468-2-14","article-title":"Subadditivity of chance measure","volume":"2","author":"Hou","year":"2014","journal-title":"J. Uncertain. Anal. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s10700-016-9242-z","article-title":"On the convergence of uncertain random sequences","volume":"16","author":"Ahmadzade","year":"2017","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_35","first-page":"31","article-title":"Further results of convergence of uncertain random sequences","volume":"15","author":"Gao","year":"2018","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_36","first-page":"325","article-title":"Sur l\u2019existence des suites de variables al\u00e9atoires s \u00e0 s ind\u00e9pendantes \u00e9changeables ou stationnaires","volume":"31","author":"Bretagnolle","year":"1995","journal-title":"Ann. De L\u2019IHP Probab. Et Statist."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Mohammed, P.O., Guirao, J.L.G., and Hamed, Y.S. (2021). Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations. Discr. Cont. Dyn. Syst.-S.","DOI":"10.3934\/dcdss.2021083"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01013465","article-title":"An elementary proof of the strong law of large numbers","volume":"55","author":"Etemadi","year":"1981","journal-title":"Z. Wahrscheinlichkeitstheorie Und Verwandte Geb."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lo\u00e8ve, M. (1977). Probability Theory I, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4757-6288-4"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gut, A. (2013). Probability: A Graduate Course, Springer Science & Business Media. [2nd ed.].","DOI":"10.1007\/978-1-4614-4708-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2258\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:36:37Z","timestamp":1760168197000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2258"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,27]]},"references-count":40,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122258"],"URL":"https:\/\/doi.org\/10.3390\/sym13122258","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,27]]}}}