Fastest convergence of sum of uniform independent distributions to a Gaussian The sum of uniform i.i.d. random variables follows the Irwin-Hall distribution. Through observation it seems that the convergence is faster in comparison to the sum of uniform independent but not identically distributed random variables. 
Is there any result that proves this conjecture?
 A: Let $U_1,U_2,\dots$ be iid rv's uniformly distributed on $[-1,1]$. If a natural number $n$ and real $a_1,\dots,a_n$ vary so that 
\begin{equation}
 \sum_1^n a_i^2=3\quad \text{and}\quad \max_1^n|a_i|\to0\tag{1}
\end{equation}
(whence $n\to\infty$), then (say) by the Berry--Esseen inequality, 
\begin{equation*}
 S_n:=\sum_1^n a_i U_i
\end{equation*}
converges to a standard normal rv $Z$ in distribution. 
The closeness of the distribution of $S_n$ to normality can be reasonably measured in an infinite variety of ways. One of them, in view of the Esseen smoothing inequality (say; see e.g. Theorem 2.5.2, page 21) is to consider the closeness of the characteristic function (cf) $f_n$ of $S_n$ to the cf $f$ of $Z$ in a neighborhood of $0$. Given (1), we have
\begin{equation}
 \ln f_n(t)-\ln f(t)=\sum_1^n\ln\frac{\sin a_i t}{a_i t}-\frac{-t^2}2
 \sim-\frac{t^4}{180}\,\sum_1^n a_i^4
\end{equation}
uniformly over all $t$ in any given neighborhood of $0$; 
here we use the asymptotic expansion 
$$\ln\frac{\sin a t}{a t}-\Big(\frac{-a^2 t^2}6\Big)\sim -\frac{a^4t^4}{180}$$ 
for $a\to0$ and $t$ in any given neighborhood of $0$. 
So, the closeness of the distribution of $S_n$ to normality can be measured by $\sum_1^n a_i^4$, which attains its minimum given the first condition in (1) when the $a_i^2$'s are the same for all $i=1,\dots,n$, that is, when the rv's $a_iU_i$ are identically distributed. 
