New Vapnik–Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The constants are explicit, enabling numerical comparisons.

中文翻译：

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罕见事件频率的统一浓度界限

为独立随机样本的经验分布推导出新的 Vapnik-Chervonenkis 型浓度不等式。重点是低概率区域内 Borel 集类别的最大偏差。常量是显式的，可以进行数值比较。