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The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality

Let F^ n 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 sup x (F^ n (x)−F(x))>λ)≤C exp(−2λ 2 ), where C is some unspecified constant. We show that C can be taken as 1, provided that exp(−2λ 2 )≤1/2. In particular, the two-sided inequality P(√n sup x |F^ n (x)−F(x)|>λ)≤2 exp(−2λ 2 ) holds without any restriction on λ. In the one-sided as well as in the two-sided case, the constants cannot be further improved