Berry–Esséen theorem

The central limit theorem in probability theory and statistics states that under certain circumstances the sample mean, considered as a random quantity, becomes more normally distributed as the sample size is increased. The Berry–Esséen theorem, also known as the Berry–Esséen inequality, attempts to quantify the rate at which this convergence to normality takes place.

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

One version, sacrificing generality somewhat for the sake of clarity, is the following:


 * Let X1, X2, ..., be i.i.d. random variables with E(X1) = 0, E(X12) = &sigma;2 > 0, and E(|X1|3) = &rho; < $$\infty$$. Also, let
 * $$Y_n = {X_1 + X_2 + \ldots + X_n \over n}$$
 * be the sample mean, with Fn the cdf of
 * $${Y_n \over {\sigma \sqrt{n}}},$$
 * and &Phi; the cdf of the standard normal distribution. Then there exists a positive constant C such that for all x and n,
 * $$\left|F_n(x) - \Phi(x)\right| \le {C \rho \over \sigma^3\,\sqrt{n}}.$$

That is: given a sequence of Independent identically-distributed random variables, each having mean zero and positive variance, if additionally the third absolute moment is finite, then the cumulative distribution functions of the standardized sample mean and the standard normal distribution differ (vertically, on a graph) by no more than the specified amount. Note that the rate of convergence is on the order of n&minus;1/2.

Calculated values of the constant C have decreased markedly over the years, from 7.59 (Esséen's original bound) to 0.7975 in 1972 (by P. van Beeck). The best current bound is 0.7655 (by I. S. Shiganov in 1986).