# American Institute of Mathematical Sciences

December  2021, 6(4): 409-460. doi: 10.3934/puqr.2021020

## On the laws of the iterated logarithm under sub-linear expectations

 1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China

Received  March 03, 2021 Accepted  November 11, 2021 Published  December 2021

Fund Project: Thanks to Professor Mingshang Hu for the constructive discussion which improved our original manuscript and the revision. Special thanks to the anonymous referees for carefully reading the manuscript and constructive comments. An example given by the referees led us to consider carefully the relationship between the capacity ${\widehat{\mathbb V}^{\ast}}$ and the probability measure, and the properties of ${\widehat{\mathbb V}^{\ast}}$. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11731012, 12031005), Ten Thousand Talents Plan of Zhejiang Province (Grant No. 2018R52042), Natural Science Foundation of Zhejiang Province (Grant No. LZ21A010002), and the Fundamental Research Funds for the Central Universities.

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.

Citation: Li-Xin Zhang. On the laws of the iterated logarithm under sub-linear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 409-460. doi: 10.3934/puqr.2021020
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