Editing Berry–Esseen theorem (section)

{{Short description|Theorem in probability theory}}
In [[probability theory]], the [[central limit theorem]] states that, under certain circumstances, the [[probability distribution]] of the scaled [[sample mean|mean of a random sample]] [[convergence in distribution|converges]] to a [[normal distribution]] as the sample size increases to infinity. 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 theory|approximation]] between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the [[Kolmogorov–Smirnov test#Kolmogorov–Smirnov statistic|Kolmogorov–Smirnov distance]]. In the case of [[Independence (probability theory)|independent samples]], the convergence rate is {{math|''n''<sup>−1/2</sup>}}, where {{math|''n''}} is the sample size, and the constant is estimated in terms of the [[Skewness|third]] absolute [[Moment (mathematics)|normalized moment]]. It is also possible to give non-uniform bounds which become more strict for more extreme events.