Theorem in probability theory
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 "n"−1/2, where "n" is the sample size, and the constant is estimated in terms of the third absolute normalized moment.
Statement of the theorem.
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.
Identically distributed summands.
One version, sacrificing generality somewhat for the sake of clarity, is the following:
There exists a positive constant "C" such that if "X"1, "X"2, ..., are i.i.d. random variables with E("X"1) = 0, E("X"12) = σ2 ; 0, and E(|"X"1|3) = ρ ; ∞, and if we define
formula_1
the sample mean, with "F""n" the cumulative distribution function of
formula_2
and Φ the cumulative distribution function of the standard normal distribution, then for all "x" and "n",
formula_3
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 (vertically, on a graph) by no more than the specified amount. 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. The estimate "C"  ; 0.4748 follows from the inequality
formula_4
since σ3 ≤ ρ and 0.33554 · 1.415  ; 0.4748. However, if ρ ≥ 1.286σ3, then the estimate 
formula_5
is even tighter.
 proved that the constant also satisfies the lower bound
 formula_6
Let "X"1, "X"2, ..., be independent random variables with E("X""i") = 0, E("X""i"2) = σ"i"2 ; 0, and E(|"X""i"|3) = ρ"i" ; ∞. Also, let
formula_7
be the normalized "n"-th partial sum. Denote "F""n" the cdf of "S""n", and Φ the cdf of the standard normal distribution. For the sake of convenience denote 
formula_8
In 1941, Andrew C. Berry proved that for all "n" there exists an absolute constant "C"1 such that
formula_9
where
formula_10
Independently, in 1942, Carl-Gustav Esseen proved that for all "n" there exists an absolute constant "C"0 such that
formula_11
where
formula_12
Non-identically distributed summands.
It is easy to make sure that ψ0≤ψ1. Due to this circumstance inequality (3) 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 "X"1, ..., "X""n" have identical distributions 
formula_13
and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.
Regarding "C"0, obviously, the lower bound established by remains valid:
 formula_14
The lower bound is exactly reached only for certain Bernoulli distributions (see for their explicit expressions).
The upper bounds for "C"0 were subsequently lowered from Esseen's original estimate 7.59 to 0.5600.
Sum of a random number of random variables.
Berry–Esseen theorems exist for the sum of a random number of random variables. The following is Theorem 1 from Korolev (1989), substituting in the constants from Remark 3. It is only a portion of the results that they established:
Let formula_15 be independent, identically distributed random variables with formula_16, formula_17, formula_18. Let formula_19 be a non-negative integer-valued random variable, independent from formula_15. Let formula_21, and define
formula_22
Then
formula_23
Multidimensional version.
As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.
Let formula_24 be independent formula_25-valued random vectors each having mean zero. Write formula_26 and assume formula_27 is invertible. Let formula_28 be a formula_29-dimensional Gaussian with the same mean and covariance matrix as formula_30. Then for all convex sets formula_31,
formula_32,
where formula_33 is a universal constant and formula_34 (the third power of the L2 norm).
The dependency on formula_35 is conjectured to be optimal, but might not be.
References.
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Bibliography.
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