Bounds on the constant in the mean central limit theorem

Abstract

Let X₁, …, X_n be independent with zero means, finite variances σ₁², …, σ_n² and finite absolute third moments. Let F_n be the distribution function of (X₁ + ⋯ + X_n)/σ, where σ² = ∑_i=1ⁿσ_i², and Φ that of the standard normal. The L¹-distance between F_n 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, …, X_n 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.