# 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?

• Intuitively varying location does not matter here as you will subtract it out, but varying range does matter, and the distribution of your sum of uniform independent random variables will be more affected by those with wide range/high variance/low density than by the others with narrow range/low variance/high density. But if they are i.i.d. there are no special wide cases to slow the convergence. Commented Jun 6, 2018 at 16:36
• Agree with Henry and extending the point: the extreme cases are (a) i.i.d. variables, (b) a single uniform variable with the same aggregate mean and variance (with the other $n-1$ variables being uniform on $[0,0]$). Of course case (b) is much farther from Gaussian.
– usul
Commented Jun 6, 2018 at 17:41

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 $$\sum_1^n a_i^2=3\quad \text{and}\quad \max_1^n|a_i|\to0\tag{1}$$ (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 $$\ln f_n(t)-\ln f(t)=\sum_1^n\ln\frac{\sin a_i t}{a_i t}-\frac{-t^2}2 \sim-\frac{t^4}{180}\,\sum_1^n a_i^4$$ 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.