[1] |
Peng, S., G-expectation, G-brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, 2, 4, 541-567 (2007) · Zbl 1131.60057 · doi:10.1007/978-3-540-70847-6_25 |
[2] |
Peng, S. G., Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010), https://arxiv.org/abs/1002.4546 |
[3] |
Zhang, L. X., The convergence of the sums of independent random variables under the sub-linear expectations, Acta Mathematica Sinica, English Series, 36, 3, 224-244 (2020) · Zbl 1434.60102 · doi:10.1007/s10114-020-8508-0 |
[4] |
Zhang, L. X., Lindeberg’s central limit theorems for martingale like sequences under sub-linear expectations, Science China Mathematics (2020) · Zbl 1480.60059 · doi:10.1007/s11425-018-9556-7 |
[5] |
Zhang, L. X., Heyde’s theorem under the sub-linear expectations, Statistics & Probability Letters, 170 (2020) · Zbl 1457.60050 · doi:10.1016/j.spl.2020.108987 |
[6] |
Liu, W.; Zhang, Y., The law of the iterated logarithm for linear processes generated by a sequence of stationary independent random variables under the sub-linear expectation, Entropy, 23 (2021) · doi:10.3390/e23101313 |
[7] |
Liu, W.; Zhang, Y., Central limit theorem for linear processes generated by iid random varialbes under the sub-linear expectation, Applied Mathematics-A Journal of Chinese Universities Series B, 36, 2, 243-255 (2015) · Zbl 1488.60033 |
[8] |
Gao, F. Q.; Xu, M. Z., Large deviations and moderate deviations for independent random variables under sublinear expectations (in Chinese), Science China Mathematics, 41, 4, 337-352 (2011) · Zbl 1488.60043 |
[9] |
Zhang, L.-X., Donsker’s invariance principle under the sub-linear expectation with an application to chung’s law of the iterated logarithm, Communications in Mathematics and Statistics, 3, 2, 187-214 (2015) · Zbl 1321.60067 · doi:10.1007/s40304-015-0055-0 |
[10] |
Zhang, L., Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Science China Mathematics, 59, 12, 2503-2526 (2016a) · Zbl 1362.60031 · doi:10.1007/s11425-016-0079-1 |
[11] |
Zhang, L. X., Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics, 59, 4, 759-768 (2016b) · Zbl 1338.60095 · doi:10.1007/s11425-015-5105-2 |
[12] |
Zhang, L. X., Strong Limit Theorems for Extended Independent and Extended Negatively Random Variables under Non-linear Expectations (2016), https://arxiv.org/abs/1608.0071v1 |
[13] |
Wu, Q., Precise asymptotics for complete integral convergence under sublinear expectations, Mathematical Problems in Engineering, 2020, 1-13 (2020) · Zbl 07347921 · doi:10.1155/2020/3145935 |
[14] |
Xu, M.; Cheng, K., Precise asymptotics in the law of the iterated logarithm under sublinear expectations, Mathematical Problems in Engineering, 2021, 1-9 (2021a) · Zbl 1512.60022 · doi:10.1155/2021/6691857 |
[15] |
Xu, M. Z.; Cheng, K., Convergence for sums of i.i.d. random variables under sublinear expectations, Journal of Inequalities and Applications, 2021 (2021b) · Zbl 1504.60052 · doi:10.1186/s13660-021-02692-x |
[16] |
Xu, M. Z.; Cheng, K., Equivalent conditions of complete \(p\)-th moment convergence for weighted sums of i. i. d. random variables under sublinear expectations, Discrete Dynamics in Nature and Society, 2021 (2021c) · Zbl 1486.60060 · doi:10.1155/2021/7471550 |
[17] |
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, Journal of Inequalities and Applications, 2017 (2017) · Zbl 1386.60117 · doi:10.1186/s13660-017-1538-1 |
[18] |
Xu, J. P.; Zhang, L. X., Three series theorem for independent random variables under sub-linear expectations with applications, Acta Mathematica Sinica, English Series, 35, 2, 172-184 (2019) · Zbl 1411.60046 · doi:10.1007/s10114-018-7508-9 |
[19] |
Xu, J.-p.; Zhang, L.-x., The law of logarithm for arrays of random variables under sub-linear expectations, Acta Mathematicae Applicatae Sinica, English Series, 36, 3, 670-688 (2020) · Zbl 1459.60073 · doi:10.1007/s10255-020-0958-8 |
[20] |
Wu, Q.; Jiang, Y., Strong law of large numbers and Chover’s law of the iterated logarithm under sub-linear expectations, Journal of Mathematical Analysis and Applications, 460, 1, 252-270 (2018) · Zbl 1380.60039 · doi:10.1016/j.jmaa.2017.11.053 |
[21] |
Chen, Z., Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59, 5, 945-954 (2016) · Zbl 1341.60015 · doi:10.1007/s11425-015-5095-0 |
[22] |
Fang, X.; Peng, S. G.; Shao, Q. M.; Song, Y. S., Limit Theorems with Rate of Convergence under Sublinear Expectations (2018), https://arxiv.org/abs/1711.10649v2 |
[23] |
Hu, F.; Chen, Z. J.; Zhang, D. F., How big are the increments of \(G\)-Brownian motion, Science China Mathematics, 57, 8, 1686-1700 (2014) · Zbl 1301.60068 · doi:10.1007/s11425-014-4816-0 |
[24] |
Hu, Z.-C.; Yang, Y.-Z., Some inequalities and limit theorems under sublinear expectations, Acta Mathematicae Applicatae Sinica, English Series, 33, 2, 451-462 (2017) · Zbl 1370.60054 · doi:10.1007/s10255-017-0673-2 |
[25] |
Kuczmaszewska, A., Complete convergence for widely acceptable random variables under the sublinear expectations, Journal of Mathematical Analysis and Applications, 484, 1 (2020) · Zbl 1471.60041 · doi:10.1016/j.jmaa.2019.123662 |
[26] |
Ding, X., A general form for precise asymptotics for complete convergence under sublinear expectation, AIMS Mathematics, 7, 2, 1664-1677 (2021) |
[27] |
Gut, A.; Spătaru, A., Precise asymptotics in the law of the iterated logarithm, Annals of Probability, 28, 4, 1870-1883 (2000) · Zbl 1044.60024 · doi:10.1214/aop/1019160511 |
[28] |
Zhang, L. X., Precise Rates in the Law of the Iterated Logarithm (2006), https://arxiv.org/abs/0610519v1 |
[29] |
Huang, W.; Zhang, L.; Jiang, Y., Precise rate in the law of iterated logarithm for \(\rho \)-mixing sequence, Applied Mathematics-A Journal of Chinese Universities Series B, 18, 4, 482-488 (2003) · Zbl 1053.60025 |
[30] |
Jiang, C.; Yang, X., Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables, Applied Mathematics-A Journal of Chinese Universities, 22, 1, 87-94 (2007) · Zbl 1125.60028 · doi:10.1007/s11766-007-0011-1 |
[31] |
Wu, H.; Wen, J., Precise rates in the law of the iterated logarithm for R/S statistics, Applied Mathematics-A Journal of Chinese Universities, 21, 4, 461-466 (2006) · Zbl 1204.60027 · doi:10.1007/s11766-006-0010-7 |
[32] |
Xiao, X.-Y.; Zhang, L.-X.; Yin, H.-W., Precise rates in the generalized law of the iterated logarithm, Statistics & Probability Letters, 83, 2, 616-623 (2013) · Zbl 1269.60038 · doi:10.1016/j.spl.2012.11.005 |
[33] |
Xiao, X.-Y.; Yin, H.-W., Precise rates in the law of iterated logarithm for the moment convergence of φ-mixing sequences, Mathematica Slovaca, 65, 6, 1571-1588 (2015) · Zbl 1389.60045 · doi:10.1515/ms-2015-0107 |
[34] |
Xu, M.; Ding, Y.; Zhou, Y., Precise rates in the generalized law of the iterated logarithm in \(\mathbb{R}^m\), Journal of Mathematical Research with Applications, 38, 1, 103-110 (2018) · Zbl 1413.60025 |
[35] |
Xu, M. Z.; Cheng, K., Precise Rates in the Generalized Law of the Iterated Logarithm in the Hilbert Space, Chinese Journal of Applied Probability and Statistics (2020), preprint |
[36] |
Xu, M. Z.; Cheng, K., Precise rates in the generalized law of the iterated logarithm in the multidimensional space, Chinese Journal of Applied Probability and Statistics, 37, 5, 507-514 (2021) · Zbl 1487.60067 |