Let $X_1,\...,X_n$ be independent with zero means, finite variances $\sigma_1^2,\...,\sigma_n^2$ and finite absolute third moments. Let $F_n$ be the distribution function of $(X_1+\...+X_n)/\sigma$, where $\sigma^2=\sum_{ i=1}^n\sigma_i^2$, and $\Phi$ that of the standard normal. The $L^1$-distance between $F_n$ and $\Phi$ then satisfies \[\Vert F_n-\Phi\Vert_1\le\frac{ 1}{ \sigma^3}\sum_{ i=1}^nE|X_i|^3.\] In particular, when $X_1,\...,X_n$ are identically distributed with variance $\sigma^2$, we have \[\Vert F_n-\Phi\Vert_1\le\frac{ E|X_1|^3}{ \sigma^3\sqrt{ n}}\qquad for all $n\in\mathbb{ N}$,\] corresponding to an $L^1$-Berry--Esseen constant of 1.
Categories: math.PR