Bounds on the constant in the mean central limit theorem

Abstract

Let $X_1,\dot{.}.,X_n$ be independent with zero means, finite variances ${\sigma }_1^2,\dot{.}.,{\sigma }_n^2$ and finite absolute third moments. Let $F_n$ be the distribution function of $(X_1+\dot{.}.+X_n)/\sigma$, 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 \[\Vert F_n-Φ\Vert_1\le\frac{1}{σ^3}\sum_{i=1}^nE|X_i|^3.\] In particular, when $X_1,\dot{.}.,X_n$ are identically distributed with variance ${\sigma }^2$, we have \[\Vert F_n-Φ\Vert_1\le\frac{E|X_1|^3}{σ^3\sqrt{n}}\qquad for all $n\in \mathbb{N}$,\] corresponding to an $L^1$-Berry--Esseen constant of 1.

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)

References in corpus (3)

• On mean central limit theorems for stationary sequences
• New estimates of the convergence rate in the Lyapunov theorem
• Normal approximation for hierarchical structures

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