<p style='text-indent:20px;'>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.</p>

中文翻译：

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亚线性期望下的迭代对数定律

<p style='text-indent:20px;'>在本文中，我们建立了亚线性期望空间中独立随机变量的迭代对数定律的一些一般形式，其中随机变量不一定同分布. 独立随机变量的最大和的指数不等式和 Kolmogorov 的逆指数不等式被确立为显示迭代对数定律的工具。作为应用，得到了亚线性期望下独立同分布随机变量的迭代对数定律的充分必要条件。论文中还表明，如果亚线性期望空间足够丰富，它就没有连续容量。