New Vapnik–Chervonkis 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.
Lhaut, S., Sabourin, A., & Segers, J. (2021). Uniform concentration bounds for frequencies of rare events (LIDAM Discussion Paper ISBA 2021/34). https://hdl.handle.net/2078.5/112066