Bounds on the constant in the mean central limit theorem

Let $X_1,\...,X_n$ be independent with zero means, finite variances $σ_1^2,\...,σ_n^2$ and finite absolute third moments. Let $F_n$ be the distribution function of $(X_1+\...+X_n)/σ$, where $σ^2=\sum_{i=1}^nσ_i^2$, and $Φ$ that of the standard normal. The $L^1$-distance between $F_n$ and $Φ$ then satisfies $\Vert F_n-Φ\Vert_1\le\frac{1}{σ^3}\sum_{i=1}^nE|X_i|^3.$ In particular, when $X_1,\...,X_n$ are identically distributed with variance $σ^2$, 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.

No reviews yet.

No comments yet.