DKW inequality for $L^1$-norm

Question

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.

Answer 1

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$.