BerryEsseen

Berry–Esseen theorem, sometimes written BerryEsseen, is a result in probability theory that provides a quantitative refinement of the central limit theorem. It gives a uniform bound on how fast the distribution of a properly normalized sum of random variables approaches the standard normal distribution as the number of summands grows.

In the classical one-dimensional setting, let X1, X2, … be independent and identically distributed with mean μ, variance σ² > 0, and finite third absolute moment E|X1 − μ|³. Let S_n = X1 + … + Xn and F_n be the distribution function of the standardized sum (S_n − nμ)/(σ√n). Let Φ be the standard normal distribution function. Then there exists a universal constant C such that for all n ≥ 1,

sup_x |F_n(x) − Φ(x)| ≤ C · E|X1 − μ|³ / (σ³√n).

Equivalently, with β₃ = E|X1 − μ|³ / σ³, the bound is sup_x |F_n(x) − Φ(x)| ≤ C β₃ / √n. The constant C is universal, and the best known absolute bounds place it around a fraction of unity (approximately 0.47 in refined results), though exact optimal values depend on the formulation.

Extensions of the theorem cover independent but non-identically distributed variables under the Lindeberg condition, and there are multivariate and other refinements. Berry–Esseen is widely used to quantify the accuracy of normal approximations for finite samples in statistics and probability.

The result is named after Andrew C. Berry and Carl-Gustav Esseen, who established initial quantitative refinements in the 1940s.

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