# American Institute of Mathematical Sciences

June  2022, 7(2): 85-100. doi: 10.3934/puqr.2022006

## 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

Received  February 20, 2022 Accepted  April 18, 2022 Published  June 2022 Early access  May 2022

Fund Project: This work was supported by the NSF of China (Grant Nos. 11731012 and 12031005), Ten Thousands Talents Plan of Zhejiang Province (Grant No. 2018R52042), NSF of Zhejiang Province (Grant No. LZ21A010002), and the Fundamental Research Funds for the Central Universities.

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.

Citation: Li-Xin Zhang. A note on the cluster set of the law of the iterated logarithm under sub-linear expectations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 85-100. doi: 10.3934/puqr.2022006
##### 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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