Abstract
In this article, we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. Our results of strong limit theorems are more general than some related results previously obtained by Thrum (1987), Li et al. (1995) and Wu (2010) in classical probability space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen, P.Y., Sung, S.H. Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hungar., 148(1): 83–95 (2016)
Chen, Y.Q., Chen, A.Y., Kai, W.NG. The strong law of large numbers for extended negatively dependent random variables. J. Appl. Prob., 4: 908–922 (2016)
Chen, Z.J. Strong laws of large numbers for sub-linear expectations. Sci. China Math., 59(5): 945–954 (2016)
Chen, Z.J., Hu, F. A law of the iterated logarithm for sublinear expectations. J. Financ Eng., 1: 1–15 (2014)
Denis, L., Martini, C. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab., 16(2): 827–852 (2006)
Feng, F.X., Wang, D.C., Wu, Q.Y. Complete and complete moment convergence for weighted sums of \(\tilde{\rho}\)-mixing random variables. J. Math. Inequal., 12(1): 201–217 (2018)
Feng, F.X., Wang, D.C., Wu, Q.Y. Complete convergence for weighted sums of negatively dependent random variables under the sub-linear expectations. Commun. Stat-Theor. M., 48(6): 1351C–1366 (2019)
Gilboa, I. Expected utility theory with purely subjective non-additive probabilities. J. Math. Econom., 16: 65–68 (1987)
Hu, C. A strong law of large numbers for sub-linear expectation under a general moment condition. Statist. Probab. Lett., 119: 248–258 (2016)
Hu, D., Chen, P.Y., Sung, S.H. Strong laws for weighted sums of ψ-mixing random variables and applications in errors-in-variables regression models. TEST., 26: 600–617 (2017)
Jing, B.Y., Liang, H.Y. 2008. Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab., 21: 890–909 (2008)
Li, D.L., Rao, M.B., Jiang, T.F., et al. 1995. Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab., 8: 49–76 (1995)
Marinacci, M. Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory., 84: 145–195 (1999)
Peligrad, M., Gut, A. Almost sure results for a class of dependent random variables. J. Theoret. Probab., 12: 87–104 (1999)
Peng, S. BSDE and related g-expectation. Pitman. Res. Notes. Math. Ser., 364: 141–159 (1997)
Peng, S. 1999. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type. Probab. Theory., 113: 473–499 (1999)
Peng, S. G-expectation, G-Brownian motion and related stochastic calculus of Ito type, In: Proceedings of the 2005 Abel Symposium. Springer, Berlin, Heidelberg, 2006
Peng, S. Law of large numbers and central limit theorem under nonlinear expectations. ArX-iv:math.PR/0702358vl [math.PR] (2007)
Peng, S. Multi-dimensional G–Brownian motion and related stochastic calculus under G-expectation. Stoch. Proc. Appl., 118(12): 2223–2253 (2008)
Peng, S. A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1 [math.PR] (2008)
Peng, S. 2010. Nonlinear Expectations and Stochastic Calculus under Uncertainty. ArXiv:1002.4546 [math.PR](2010)
Shen, A.T., Xue, M.X., Wang, W.J. Complete convergence for weighted sums of extended negatively dependent random variables. Commun Stat-Theor. M., 46(3): 1433–1444 (2017)
Sung, S.H. 2011. On the strong convergence for weighted sums of random variables. Stat. Papers, 52: 447–454 (2011)
Thrum, R. A remark on almost sure convergence of weighted sums. Probab. Theory Rel., 75: 425–430 (1987)
Wu, Q.Y. 2010. Complete convergence for negatively dependent sequences of random variables. J. Inequal. Appl., Article ID 507293 10 pages (2010)
Wu, Q.Y., Jiang, Y.Y. Strong law of large numbers and Chover’s law of the iterated logarithm under sub-linear expectations. J. Math. Anal. Appl., 460: 252–270 (2018)
Zhang, L.X. Donsker’s invariance principle under the sub-linear expectation with an application to Chungs law of the iterated logarithm. Commun. Math. Stat., 3: 187–214 (2015)
Zhang, L.X. Exponential inequalities under sub-linear expectations with applications to laws of the iterated logarithm. Sci. China Math., 59(12): 2503–2526 (2016)
Zhang, L.X. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math., 59(4): 751–768 (2016)
Zhang, L.X. Strong limit theorems for extended independent and extended neg- atively dependent random variables under non-linear expectations. Acta. Math. Sci., 42(2): 467–490 (2022)
Zhong, H.Y., Wu, Q.Y. Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation. J. Inequal. Appl., 261: DOI:https://doi.org/10.1186/s13660-017-1538-1 (2017)
Acknowledgments
The authors would like to thank the editors and the reviewers for their careful reading our paper and constructive comments, which have led to significant improvements of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflict of interest.
Additional information
This research is supported by the Natural Science Foundation of Guangxi (Grant No.2024GXNSFAA010476) and the National Natural Science Foundation of China (Grant No.12361031).
Rights and permissions
About this article
Cite this article
Feng, Fx., Wang, Dc., Wu, Qy. et al. Strong Limit Theorems for Weighted Sums under the Sub-linear Expectations. Acta Math. Appl. Sin. Engl. Ser. 40, 862–874 (2024). https://doi.org/10.1007/s10255-024-1127-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-024-1127-2