# Bounds on the constant in the mean central limit theorem

Research paper by **Larry Goldstein**

Indexed on: **19 Oct '10**Published on: **19 Oct '10**Published in: **Mathematics - Probability**

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#### Abstract

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.