In this paper, we obtain a Bernstein-type concentration inequality and McDiarmid’s inequality under upper probabilities for exponential independent random variables. Compared with the classical result, our inequalities are investigated under a family of probability measures, rather than one probability measure. As applications, the convergence rates of the law of large numbers and the Marcinkiewicz–Zygmund-type law of large numbers about the random variables in upper expectation spaces are obtained.

中文翻译：

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高概率的浓度不等式

在本文中，我们在指数独立随机变量的较高概率下获得了Bernstein型浓度不等式和McDiarmid不等式。与经典结果相比，我们的不等式是根据一系列概率测度而不是一种概率测度进行研究的。作为应用，获得了关于高期望空间中随机变量的大数定律和大数Marcinkiewicz-Zygmund型定律的收敛速度。