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
$\Vert F_{n}-\Phi\Vert_{1}\le\frac{1}{\sigma^{3}}\sum_{i=1}^{n}E|X_{i}|^{3}.$
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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,
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corresponding to an L1-Berry–Esseen constant of 1.