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I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.

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    $\begingroup$ What part of the statement of the inequality is raising doubts as to its validity in the discrete case? $\endgroup$
    – cardinal
    Jun 1 '15 at 17:19
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Yes, it does. See Theorem 11.6 in Kosorok (2008) Introduction to Empirical Processes and Semiparametric Inference:

Theorem: for any iid sample $X_1,\dots,X_n$ with distribution $F$, $$\mathrm{P}\left(\sup_{t\in\mathbb{R}} \sqrt{n}\, \vert \mathbb{F}_n(t) - F(t)\vert > x\right) \leq 2 \exp(-2x^2),$$ for all $x>0$.

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