Open Access
July 2010 Bounds on the constant in the mean central limit theorem
Larry Goldstein
Ann. Probab. 38(4): 1672-1689 (July 2010). DOI: 10.1214/10-AOP527


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 $$\Vert F_{n}-\Phi\Vert_{1}\le\frac{E|X_{1}|^{3}}{\sigma^{3}\sqrt{n}} \text{for all } n ∈ ℕ,$$ corresponding to an L1-Berry–Esseen constant of 1.


Download Citation

Larry Goldstein. "Bounds on the constant in the mean central limit theorem." Ann. Probab. 38 (4) 1672 - 1689, July 2010.


Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1195.60034
MathSciNet: MR2663641
Digital Object Identifier: 10.1214/10-AOP527

Primary: 60F05 , 60F25

Keywords: Berry–Esseen constant , Stein’s method

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 4 • July 2010
Back to Top