Berry-Esseen Theorems. Please help

Question

Suppose $X_i\sim U[0,1]$. Is it true that, for some finite constant $c$, $$\left|\mathbb P\left(\sqrt{12n}\left(\bar{X} - \frac12\right)\le x\right) - \Phi(x)\right|\le\frac cn$$ uniformly in x?

Answer 1

Yes, but not because of the Berry-Esseen theorem. That result gives a uniform bound of order $O(1/\sqrt n)$. Your inequality follows from the Edgeworth expansion (see, eg, Limit distributions for sums of independent random variables by Gnedenko and Kolmogorov, section 45, or An introduction to probability theory and its applications, vol 2, by Feller, chapter 16). The idea is that the correct cdf is approximated by the gaussian cdf plus a certain function of $x$ divided by $\sqrt n$, plus an error uniformly bounded by $O(1/n)$. In your special case, the $1/\sqrt n$ term vanishes.