The Annals of Probability

Bounds on the constant in the mean central limit theorem

Larry Goldstein

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Let X1, …, Xn be independent with zero means, finite variances σ12, …, σn2 and finite absolute third moments. Let Fn be the distribution function of (X1 + ⋯ + Xn)/σ, where σ2 = ∑i=1nσi2, and Φ that of the standard normal. The L1-distance between Fn and Φ then satisfies

$\Vert F_{n}-\Phi\Vert_{1}\le\frac{1}{\sigma^{3}}\sum_{i=1}^{n}E|X_{i}|^{3}.$

In particular, when X1, …, Xn are identically distributed with variance σ2, we have

$\Vert F_{n}-\Phi\Vert_{1}\le\frac{E|X_{1}|^{3}}{\sigma^{3}\sqrt{n}}$  for all n ∈ ℕ,

corresponding to an L1-Berry–Esseen constant of 1.

Article information

Ann. Probab., Volume 38, Number 4 (2010), 1672-1689.

First available in Project Euclid: 8 July 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F25: $L^p$-limit theorems

Stein’s method Berry–Esseen constant


Goldstein, Larry. Bounds on the constant in the mean central limit theorem. Ann. Probab. 38 (2010), no. 4, 1672--1689. doi:10.1214/10-AOP527.

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