Berry–Esseen theorem

The Berry–Esseen theorem bounds the maximum difference between a standardized sum’s cumulative distribution function and the standard normal distribution function, using the summands’ third absolute moments.

Kolmogorov distance: The maximum absolute difference between two cumulative distribution functions. This is the distance Berry–Esseen bounds between the sum and the normal law.
Andrew C. Berry: A British mathematician whose work on rates of convergence in the central limit theorem appeared in 1941. His independent proof gave the result its Berry half of the name.
Normal approximation: The replacement of a probability distribution or statistic by a normal distribution for calculation. Berry–Esseen supplies a rigorous upper bound on the resulting cumulative-probability error.
Lindeberg–Feller central limit theorem: A central limit theorem for triangular arrays under a condition limiting the influence of large summands. It provides broad convergence conditions, while Berry–Esseen gives a rate under stronger moment control.
Third absolute moment: The expectation of the absolute value of a random variable cubed, when finite. The theorem’s error bound grows with this measure of each summand’s tail size.
Carl-Gustav Esseen: A Swedish mathematician who developed foundational methods for bounding central-limit approximation errors. His 1942 work established a central version of the theorem and introduced Fourier-analytic techniques.
Binomial distribution: The distribution of the number of successes in a fixed number of independent trials with equal success probability. Its standardized form can be approximated by a normal law, with error bounded using the theorem.
Lyapunov central limit theorem: A central limit theorem for independent sums under a condition on moments of order greater than two. It guarantees convergence but does not itself give Berry–Esseen’s explicit uniform error rate.
Standardization: A transformation that subtracts a distribution’s mean and divides by its standard deviation. The bound compares the standardized sum with a normal distribution of mean zero and variance one.
Characteristic function: The expected value of the complex exponential of a random variable, which determines its probability distribution. Esseen’s approach uses characteristic functions to translate moment information into distributional error bounds.

Linked from 2 pages

De Moivre–Laplace theoremRelated: It quantifies finite-sample error beyond the qualitative convergence in De Moivre–Laplace.