Bounds on the constant in the mean central limit theorem

Details Bounds on the constant in the mean central limit theorem Bounds on the constant in the mean central limit theorem

2009-06-28

arxiv.org/abs/0906.5145v2

Annals of Probability 2010, Vol. 38, No. 4, 1672-1689

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP527 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP527

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.

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