Berry–Esseen theorem

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.

Statements of the theorem vary, as it was independently discovered by two mathematicians, Andrew C. Berry (in 1941) and Carl-Gustav Esseen (1942), who then, along with other authors, refined it repeatedly over subsequent decades.

Note that the approximation error for all n (and hence the limiting rate of convergence for indefinite n sufficiently large) is bounded by the order of n−1/2.

Calculated upper bounds on the constant C have decreased markedly over the years, from the original value of 7.59 by Esseen in 1942.

Moreover, in the case where the summands X1, ..., Xn have identical distributions and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.