The Berry-Esseen theorem is named after him.

More generally, the rate of convergence can be roughly quantified by e.g. Chebyshev's inequality and the Berry-Esseen theorem.

Esseen's bound is now called "the Berry-Esseen theorem", because it was independently proved by Andrew C. Berry, also.

A general upper bound for the approximation error in the central limit theorem is given by the Berry-Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

Under stronger assumptions, the Berry-Esseen theorem, or Berry-Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean.

With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10,000 independent samples the normal distribution approximates (to two digits of accuracy) the distribution of the sample mean for many population distributions, by the Berry-Esseen theorem.