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A note on the cluster set of the law of the iterated logarithm under sub-linear expectations
1. | School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China |
In this note, we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space. For showing the result, a self-normalized law of the iterated logarithm is established.
References:
[1] |
Hu, M., Li, X. and Li, X., Convergence rate of Peng’s law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(3): 261−266.
doi: 10.3934/puqr.2021013. |
[2] |
Guo, X., Li, S. and Li, X., On the laws of the iterated logarithm with mean uncertainty under the sublinear expectation, Probab. Uncertain. Quant. Risk, 2022, 7(1): 1−22.
doi: 10.3934/puqr.2022001. |
[3] |
Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Stat. Probab. Lett., 2021, 172: 109042.
doi: 10.1016/j.spl.2021.109042. |
[4] |
de la Peña, V. H., A general class of exponential inequalities for martingales and ratios, Ann. Probab., 1999, 27: 537−564. |
[5] |
de la Peña, V. H., Lai, T. L. and Shao, Q. M., Self-Normalized Processes: Limit Theory and Statistical Applications, Springer-Verlag, Berlin Heidelberg, 2009. |
[6] |
Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. |
[7] |
Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, 2019. |
[8] |
Zhang, L. X., Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Sci. China Math., 2016a, 59(12): 2503−2526. |
[9] |
Zhang, L. X., Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation, Commun. Math. Stat., 2016b, 4: 229−263.
doi: 10.1007/s40304-015-0084-8. |
[10] |
Zhang, L. X., On the laws of the iterated logarithm under sub-linear expectations., Probab. Uncertain. Quant. Risk, 2021a, 6(4): 409−460. |
[11] |
Zhang, L. X., The sufficient and necessary conditions of the strong law of large numbers under the sub-linear expectations, arXiv: 2104.08471, 2021b. |
show all references
References:
[1] |
Hu, M., Li, X. and Li, X., Convergence rate of Peng’s law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(3): 261−266.
doi: 10.3934/puqr.2021013. |
[2] |
Guo, X., Li, S. and Li, X., On the laws of the iterated logarithm with mean uncertainty under the sublinear expectation, Probab. Uncertain. Quant. Risk, 2022, 7(1): 1−22.
doi: 10.3934/puqr.2022001. |
[3] |
Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Stat. Probab. Lett., 2021, 172: 109042.
doi: 10.1016/j.spl.2021.109042. |
[4] |
de la Peña, V. H., A general class of exponential inequalities for martingales and ratios, Ann. Probab., 1999, 27: 537−564. |
[5] |
de la Peña, V. H., Lai, T. L. and Shao, Q. M., Self-Normalized Processes: Limit Theory and Statistical Applications, Springer-Verlag, Berlin Heidelberg, 2009. |
[6] |
Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. |
[7] |
Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, 2019. |
[8] |
Zhang, L. X., Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Sci. China Math., 2016a, 59(12): 2503−2526. |
[9] |
Zhang, L. X., Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation, Commun. Math. Stat., 2016b, 4: 229−263.
doi: 10.1007/s40304-015-0084-8. |
[10] |
Zhang, L. X., On the laws of the iterated logarithm under sub-linear expectations., Probab. Uncertain. Quant. Risk, 2021a, 6(4): 409−460. |
[11] |
Zhang, L. X., The sufficient and necessary conditions of the strong law of large numbers under the sub-linear expectations, arXiv: 2104.08471, 2021b. |
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