과제정보
The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.
참고문헌
- F. Boukhari, On a Spitzer-type law of large numbers for partial sums of m-negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RAC-SAM 115 (2021), no. 4, Paper No. 180, 10 pp. https://doi.org/10.1007/s13398-021-01128-x
- Z. Chen and F. Hu, A law of the iterated logarithm under sublinear expectations, Int. J. Financ. Eng. 1 (2014), no. 2, 1450015, 23 pp. https://doi.org/10.1142/s2345768614500159
- Z. Chen, P. Y. Wu, and B. M. Li, A strong law of large numbers for non-additive probabilities, Internat. J. Approx. Reason. 54 (2013), no. 3, 365-377. https://doi.org/10.1016/j.ijar.2012.06.002
- G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131-295 (1955). https://doi.org/10.5802/aif.53
- L. Denis, M. Hu, and S. G. Peng, Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal. 34 (2011), no. 2, 139-161. https://doi.org/10.1007/s11118-010-9185-x
- L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab. 16 (2006), no. 2, 827-852. https://doi.org/10.1214/105051606000000169
- X. Feng, Law of the logarithm for weighted sums of negatively dependent random variables under sublinear expectation, Statist. Probab. Lett. 149 (2019), 132-141. https://doi.org/10.1016/j.spl.2019.01.033
- I. Gilboa, Expected utility with purely subjective nonadditive probabilities, J. Math. Econom. 16 (1987), no. 1, 65-88. https://doi.org/10.1016/0304-4068(87)90022-X
- X. Guo and X. Li, On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Statist. Probab. Lett. 172 (2021), Paper No. 109042, 8 pp. https://doi.org/10.1016/j.spl.2021.109042
- P. L. Hsu and H. E. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25
- F. Hu and Z. Chen, General laws of large numbers under sublinear expectations, Comm. Statist. Theory Methods 45 (2016), no. 14, 4215-4229. https://doi.org/10.1080/03610926.2014.917677
- T. C. Hu, C. Y. Chiang, and R. L. Taylor, On complete convergence for arrays of rowwise m-negatively associated random variables, Nonlinear Anal. 71 (2009), no. 12, e1075-e1081. https://doi.org/10.1016/j.na.2009.01.104
- Y. J. Hu, R. Ming, and W. Q. Yang, Large deviations and moderate deviations for m-negatively associated random variables, Acta Math. Sci. Ser. B (Engl. Ed.) 27 (2007), no. 4, 886-896. https://doi.org/10.1016/S0252-9602(07)60086-1
- W. Huang and P. Y. Wu, Strong laws of large numbers for general random variables in sublinear expectation spaces, J. Inequal. Appl. 2019 (2019), Paper No. 143, 18 pp. https://doi.org/10.1186/s13660-019-2094-7
- A. Kuczmaszewska, Complete convergence for widely acceptable random variables under the sublinear expectations, J. Math. Anal. Appl. 484 (2020), no. 1, 123662, 15 pp. https://doi.org/10.1016/j.jmaa.2019.123662
- Y. Lin and X. Feng, Complete convergence and strong law of large numbers for arrays of random variables under sublinear expectations, Comm. Statist. Theory Methods 49 (2020), no. 23, 5866-5882. https://doi.org/10.1080/03610926.2019.1625924
- F. Maccheroni and M. Marinacci, A strong law of large numbers for capacities, Ann. Probab. 33 (2005), no. 3, 1171-1178. https://doi.org/10.1214/009117904000001062
- S. G. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probab. Theory Related Fields 113 (1999), no. 4, 473-499. https://doi.org/10.1007/s004400050214
- S. G. Peng, G-expectation, G-Brownian motion and related stochastic calculus of ito's type, Stochastic Anal. Appl. 2 (2006), no. 4, 541-567.
- S. G. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl. 118 (2008), no. 12, 2223-2253. https://doi.org/10.1016/j.spa.2007.10.015
- S. G. Peng, Nonlinear expectations and stochastic calculus under uncertainty with robust CLT and G-Brownian motion, Springer, Berlin, Heidelberg, 2019.
- A. Shen, Y. Zhang, B. Xiao, and A. Volodin, Moment inequalities for m-negatively associated random variables and their applications, Statist. Papers 58 (2017), no. 3, 911-928. https://doi.org/10.1007/s00362-015-0731-x
- M. Wang and X. J. Wang, Some convergence properties for the maximum of partial sums of m-negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2345-2358. https://doi.org/10.1007/s13398-019-00626-3
- Y. F. Wu, T. C. Hu, and A. I. Volodin, Complete convergence and complete moment convergence for weighted sums of m-NA random variables, J. Inequal. Appl. 2015 (2015), 200, 14 pp. https://doi.org/10.1186/s13660-015-0717-1
- Q. Y. Wu and Y. Jiang, Strong law of large numbers and Chover's law of the iterated logarithm under sub-linear expectations, J. Math. Anal. Appl. 460 (2018), no. 1, 252-270. https://doi.org/10.1016/j.jmaa.2017.11.053
- M. Xu and K. Cheng, Precise asymptotics in the law of the iterated logarithm under sublinear expectations, Math. Probl. Eng. 2021 (2021), Art. ID 6691857, 9 pp. https://doi.org/10.1155/2021/6691857
- J. P. Xu and L. X. Zhang, Three series theorem for independent random variables under sub-linear expectations with applications, Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 2, 172-184. https://doi.org/10.1007/s10114-018-7508-9
- L. X. Zhang, Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Sci. China Math. 59 (2016), no. 12, 2503-2526. https://doi.org/10.1007/s11425-016-0079-1
- L. X. Zhang, Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci. China Math. 59 (2016), no. 4, 751-768. https://doi.org/10.1007/s11425-015-5105-2
- L. X. Zhang, Heyde's theorem under the sub-linear expectations, Statist. Probab. Lett. 170 (2021), Paper No. 108987, 9 pp. https://doi.org/10.1016/j.spl.2020.108987
- L. X. Zhang, On the laws of the iterated logarithm under sub-linear expectations, Probab. Uncertain. Quant. Risk 6 (2021), no. 4, 409-460. https://doi.org/10.3934/puqr.2021020
- L. X. Zhang, Strong limit theorems for extended independent random variables and extended negatively dependent random variables under sub-linear expectations, Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), no. 2, 467-490. https://doi.org/10.1007/s10473-022-0203-z
- N. Zhang and Y. T. Lan, Rosenthal's inequalities for asymptotically almost negatively associated random variables under upper expectations, Chinese Ann. Math. Ser. B 40 (2019), no. 1, 117-130. https://doi.org/10.1007/s11401-018-0122-4
- L. X. Zhang and J. Lin, Marcinkiewicz's strong law of large numbers for nonlinear expectations, Statist. Probab. Lett. 137 (2018), 269-276. https://doi.org/10.1016/j.spl.2018.01.022