# DKW inequality for $L^1$-norm

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 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.

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 $$n \to \infty$$.