# Is there an example that both Berry-Essen bound and DKW bound are attained?

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\sigma _{i}^{2}}{\Big )}^{{-3/2}}\cdot \sum \limits _{{i=1}}^{n}\rho _{i}$ and $\sigma_i,\rho_i$ are second and third moments respectively. $\widehat{F_n(x)}$ is the cumulative cdf for $\frac{X_1+\cdots+X_n}{\sqrt{\sigma^2_1+\cdots \sigma^2_n}}$

The DKW bound stated that $$\Pr {\Bigl (}\sup _{{x\in {\mathbb{R}}}}|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\leq 2e^{{-2n\varepsilon ^{2}}}\qquad {\text{for every }}\varepsilon >0$$.

Question. Is it possible to construct such an example $F(x)$ such that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\sim O(\psi_0(n))$$ and simultaneously $$\Pr {\Bigl (}\sup _{{x\in {\mathbb{R}}}}|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\sim O(e^{-2n\epsilon})$$ as $n\rightarrow \infty$.

(Note: My motivation is to find such a distribution worse enough to attain these two bounds tightly asymptotically, not to prove "$F=\Phi$" as @Linden pointed out, this may be a possible confusion of motivation.)

• Is it really the same function $F_n$ in both bounds? In the Berry-Essen bound, $$F_n(x) = P\left((\sum_{i=1}^{n} \sigma_i^2)^{-1/2} \sum_{i=1}^{n} X_i \leq x\right).$$ In the DKW bound, $$F_n(x) = n^{-1} \sum_{i=1}^{n} 1_{\{X_i \leq x\}}.$$ The second function is random. For emphasis, we can write it as $$F_n(x;\omega) = n^{-1} \sum_{i=1}^{n} 1_{\{\omega:X_i(\omega) \leq x\}}.$$ – Linden Apr 25 '17 at 22:27
• Also, for the Berry-Essen bound, the cdf of each $X_1,\ldots,X_n$ is $\Phi(x)$, the standard normal cdf. And in the DKW bound, the cdf of each $X_1,\ldots,X_n$ is $F(x)$. So do you want $F(x)=\Phi(x)$? – Linden Apr 25 '17 at 22:35
• @Linden For your first comment, they are indeed different. I edited the post to make it clear(I followed wiki's notations in the link, but it is obvious better to use different notations in one post). For your second comment, I do not see why $F=\Phi$ since the second one is a probability statement.... – Henry.L Apr 25 '17 at 22:57
• I see now. You want to find a cdf $F$ such that, given random variables $X_1,\ldots,X_n \sim F$, you have $$\sup_{x} |\widehat{F_n(x)}-\Phi(x)| \geq C_1\psi_0(n)$$ and $$P(\sup_{x} |F_n(x)-\Phi(x)| > \epsilon) \geq C_2e^{-2n\epsilon}$$ for all sufficiently large $n$. Here $$\widehat{F_n(x)} = P( \sigma_n^{-1} \sum_{i=1}^{n} X_i \leq x ), \quad \sigma_n = (\sum_{i=1}^{n} \sigma_i^2)^{1/2}$$ and $$F_n(x) = n^{-1} \sum_{i=1}^{n} 1_{X_i \leq x}.$$ – Linden Apr 26 '17 at 0:25
• What are the standard examples of cdfs $F$ that show (at least) the Berry-Essen bound is tight? What are the standard examples of cdfs $F$ that show (at least) the DKW bound is tight? I guess you've already looked those up and tried to combine them? – Linden Apr 26 '17 at 0:29