English

Bounds on the constant in the mean central limit theorem

Probability 2010-10-20 v2

Abstract

Let X1,.˙.,XnX_1,\...,X_n be independent with zero means, finite variances σ12,.˙.,σn2\sigma_1^2,\...,\sigma_n^2 and finite absolute third moments. Let FnF_n be the distribution function of (X1+.˙.+Xn)/σ(X_1+\...+X_n)/\sigma, where σ2=i=1nσi2\sigma^2=\sum_{i=1}^n\sigma_i^2, and Φ\Phi that of the standard normal. The L1L^1-distance between FnF_n and Φ\Phi then satisfies FnΦ11σ3i=1nEXi3.\Vert F_n-\Phi\Vert_1\le\frac{1}{\sigma^3}\sum_{i=1}^nE|X_i|^3. In particular, when X1,.˙.,XnX_1,\...,X_n are identically distributed with variance σ2\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 L1L^1-Berry--Esseen constant of 1.

Keywords

Cite

@article{arxiv.0906.5145,
  title  = {Bounds on the constant in the mean central limit theorem},
  author = {Larry Goldstein},
  journal= {arXiv preprint arXiv:0906.5145},
  year   = {2010}
}

Comments

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

R2 v1 2026-06-21T13:18:41.367Z