Berry–Esseen theorem


In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample 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 between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the Kolmogorov–Smirnov distance. In the case of independent samples, the convergence rate is, where is the sample size, and the constant is estimated in terms of the third absolute normalized moments.

Statement of the theorem

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

Identically distributed summands

One version, sacrificing generality somewhat for the sake of clarity, is the following:
That is: given a sequence of independent and 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 by no more than the specified amount. Note that the approximation error for all n is bounded by the order of n−1/2.
Calculated values of the constant C have decreased markedly over the years, from the original value of 7.59 by, to 0.7882 by, then 0.7655 by, then 0.7056 by, then 0.7005 by, then 0.5894 by, then 0.5129 by, then 0.4785 by. The detailed review can be found in the papers and. The best estimate, C < 0.4748, follows from the inequality
due to, since σ3 ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ3, then the estimate
which is also proved in, gives an even tighter upper estimate.
proved that the constant also satisfies the lower bound

Non-identically distributed summands

It is easy to make sure that ψ0≤ψ1. Due to this circumstance inequality is conventionally called the Berry–Esseen inequality, and the quantity ψ0 is called the Lyapunov fraction of the third order. Moreover, in the case where the summands X1,..., Xn have identical distributions
and thus the bounds stated by inequalities, and coincide apart from the constant.
Regarding C0, obviously, the lower bound established by remains valid:
The upper bounds for C0 were subsequently lowered from the original estimate 7.59 due to to 0.9051 due to, 0.7975 due to, 0.7915 due to, 0.6379 and 0.5606 due to and. the best estimate is 0.5600 obtained by.

Multidimensional version

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.
Let be independent -valued random vectors each having mean zero. Write and assume is invertible. Let be a -dimensional Gaussian with the same mean and covariance matrix as. Then for all convex sets,
where is a universal constant and .
The dependency on is conjectured to be optimal, but might not be necessary.