Abstract

In this paper, we obtain a new kind of strong law of large numbers in sublinear expectation spaces without the independent or identically distributed assumption. Instead, the strong law of large numbers is based on some restrictions on the sub-linear expectations of the partial sums. As an application, we give an interesting example in which the random variable sequence enjoys our moment restrictions, but does not enjoy the independent conditions introduced by Marinacci (1999 Marinacci, M. 1999. Limit laws for non-additive probabilities and their frequentist interpretation. Journal of Economic Theory 84 (2):145–95. doi:10.1006/jeth.1998.2479.[Crossref], [Web of Science ®] , [Google Scholar]), Peng (2009 Peng, S. 2009. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sub-linear expectations. Science in China Series A: Mathematics 52 (7):1391–411. doi:10.1007/s11425-009-0121-8.[Crossref] , [Google Scholar]), Chen, Wu, and Li (2013 Chen, Z., P. Wu, and B. Li. 2013. A strong law of large numbers for non-additive probabilities. International Journal of Approximate Reasoning 54 (3):365–77. doi:10.1016/j.ijar.2012.06.002.[Crossref], [Web of Science ®] , [Google Scholar]), Zhang (2016b Zhang, L. 2016b. Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations. ArXiv:1608.00710v1.[Crossref] , [Google Scholar]), nor Chen, Huang, and Wu (2019 Chen, Z., W. Huang, and P. Wu. 2019. Extension of the strong law of large numbers for capacities. Mathematical Control and Related Fields 9 (1):175–90. doi:10.3934/mcrf.2019010.[Crossref], [Web of Science ®] , [Google Scholar]).