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Let Fₙ denote the empirical distribution function for a sample of n i. i. d. random variables with distribution function F. In 1956 Dvoretzky, Kiefer and Wolfowitz proved that P (n ₓ (Fₙ (x) - F (x) ) >) C (-2²), where C is some unspecified constant. We show that C can be taken as 1 (as conjectured by Birnbaum and McCarty in 1958), provided that (-2²) 12. In particular, the two-sided inequality P (n ₓ|Fₙ (x) - F (x) | >) 2 (-2²) holds without any restriction on. In the one-sided as well as in the two-sided case, the constants cannot be further improved.
Pascal Massart (Sun,) studied this question.