Suppose that $X,X\_1,X\_2,X\_3\dots$ is a sequence of $\mathbb{P}$-i.i.d. random variables **supported in the interval $[0,1]$**. Let $F$ be the cumulative distribution of $X$, i.e. $F(x):=\mathbb{P}[X \le x]$ and let $\hat{F}\_n$ be the empirical cumulative distribution given $X\_1,\dots,X\_n$, i.e, $\hat{F}\_n(x) := \frac{1}{n}\sum\_{k=1}^n\mathbb{I}\{X\_k \le x\}$.


It is known from the [DKW inequality](https://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality#The_DKW_inequality) that for every $n \in \mathbb{N}$ and $\varepsilon >0$
\begin{equation\*}
 \mathbb{P}[\|\hat{F}\_n-F\|\_{L^\infty([0,1])} \ge \varepsilon ] =\mathbb{P}\bigg[\sup\_{x \in [0,1]}|\hat{F}\_n(x)-F(x)| \ge \varepsilon\bigg] \le 2 \exp(-2\cdot\varepsilon^2 \cdot n) \;.
\end{equation\*}


I'm wondering if can *strengthen* the DKW inequality if we replace the norm of $L^\infty([0,1])$ with the norm of $L^1([0,1])$. Specifically, can we find a universal constant $\alpha < 2$ and other two universal constants $c\_1>0, c\_2>0$ such that, regardless what is the distribution $F$ of $X$, it holds that for each $n \in \mathbb{N}$ and $\varepsilon >0$:
\begin{equation\*}
 \mathbb{P}[\|\hat{F}\_n-F\|\_{L^1([0,1])} \ge \varepsilon ] =\mathbb{P}\bigg[\int\_0^1|\hat{F}\_n(x)-F(x)| \mathrm{d}x \ge \varepsilon\bigg] \le c\_1 \exp(-c\_2\cdot\varepsilon^\alpha \cdot n) \;?
\end{equation\*}


I value even something not exactly like this but similar in spirit. Any pointer to the literature is very welcome.



DKW inequality for $L^1$-norm
The exponent 2 on $\epsilon$ cannot be improved by passing to the $L^1$ norm. Consider $X\_i$ i.i.d. uniform in $[0,1]$. Let $C$ be a constant, and denote $\varepsilon\_n= C/\sqrt{n}$. Let $A\_n $ be the event that $\hat{F}\_n(1/2)<1/2-9\varepsilon\_n$ and let $B\_n$ be the event that $\|\hat{F}\_n-F\|\_{L^1([0,1])} \ge \varepsilon\_n$.


Then $P(A\_n) \to \Phi(-18C)$ as $n \to \infty$ by the central limit theorem, and for large $n$ we can infer from the DKW inequality that
$P(B\_n|A\_n)>1/2$. To do that, first condition on $A\_n$, and then apply DKW separately in $[0,1/2]$ and in $[1/2,1]$.


Thus $\liminf\_n P(B\_n) >0$, but for $\alpha<2$ we have
$\exp(-c\_2\cdot\varepsilon^\alpha \cdot n) \to 0$ as $ n \to \infty$.


