On convergences of uncertain random sequences under U-S chance spaces On Convergences of Uncertain Random

• Published in: Fuzzy optimization and decision making Vol. 24; no. 3; pp. 485 - 529
• Main Authors: Yang, Deguo, Zong, Zhaojun, Hu, Feng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2025
• Subjects: Artificial Intelligence; Calculus of Variations and Optimal Control; Optimization; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Probability Theory and Stochastic Processes; U-S chance space; Uncertain random sequence; Continuity; Convergence
• ISSN: 1568-4539; 1573-2908
• DOI: 10.1007/s10700-025-09454-0

Convergence has been a topic of considerable interest. This study further develops U-S chance theory to investigate convergences for uncertain random sequences in complex systems where human uncertainty and randomness with sub-linear characteristics coexist. Building upon two existing chance measures, this paper defines two new chance measures, presents their properties and proves the relationship among the four chance measures. Six expectations of uncertain random variables under U-S chance spaces are suggested based on Choquet integrals and sub-linear expectations. Meanwhile, their relationship and Markov’s inequality are proven. Furthermore, this paper systematically presents multiple definitions of the continuities for chance measures under U-S chance spaces and investigate the relationships among them. Based on the continuity assumption of uncertain measure, a new version of Borel-Cantelli lemma under U-S chance spaces is proven. Finally, several definitions of convergences for uncertain random sequences under U-S chance spaces are presented. By rigorous mathematical proofs and systematic construction of counterexamples, the relationships among different types of convergences are illustrated.