Editing Berry–Esseen theorem (section)

===Identically distributed summands===

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

:There exists a positive [[Constant (mathematics)|constant]] ''C'' such that if ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., are [[Independent and identically distributed random variables|i.i.d. random variables]] with [[Expected value|E]](''X''<sub>1</sub>) = 0, E(''X''<sub>1</sub><sup>2</sup>) = ''σ''<sup>2</sup> > 0, and E(|''X''<sub>1</sub>|<sup>3</sup>) = ''ρ'' < ∞,<ref group="note">Since the random variables are identically distributed, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... all have the same [[moment (mathematics)|moments]] as ''X''<sub>1</sub>.</ref> and if we define
::<math>Y_n = {X_1 + X_2 + \cdots + X_n \over n}</math>
:the [[sample mean]], with ''F''<sub>''n''</sub> the [[cumulative distribution function]] of
::<math>{Y_n \sqrt{n} \over {\sigma}},</math><!-- please DO NOT CHANGE this formula unless you have read and understood the relevant comments on the talk page -->
:and Φ the cumulative distribution function of the [[standard normal distribution]], then for all ''x'' and ''n'',
::<math>\left|F_n(x) - \Phi(x)\right| \le {C \rho \over \sigma^3\sqrt{n}}.\ \ \ \ (1)</math>

[[Image:BerryEsseenTheoremCDFGraphExample.png|thumb|250px|Illustration of the difference in cumulative distribution functions alluded to in the theorem.]]
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 (mathematics)|moment]] is finite, then the [[cumulative distribution function]]s of the [[Standard score|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 [[Big O notation|order]] of ''n''<sup>−1/2</sup>.

Calculated upper bounds on the constant ''C'' have decreased markedly over the years, from the original value of 7.59 by Esseen in 1942.<ref>{{harvtxt|Esseen|1942}}. For improvements see {{harvtxt|van Beek|1972}}, {{harvtxt|Shiganov|1986}}, {{harvtxt|Shevtsova|2007}}, {{harvtxt|Shevtsova|2008}},  {{harvtxt|Tyurin|2009}}, {{harvtxt|Korolev|Shevtsova|2010a}}, {{harvtxt|Tyurin|2010}}. The detailed review can be found in the papers {{harvtxt|Korolev|Shevtsova|2010a}} and {{harvtxt|Korolev|Shevtsova|2010b}}.</ref> The estimate ''C''&nbsp;<&nbsp;0.4748 follows from the inequality
:<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le {0.33554 (\rho+0.415\sigma^3)\over \sigma^3\sqrt{n}},</math>
since ''σ''<sup>3</sup>&nbsp;≤&nbsp;''ρ'' and 0.33554&nbsp;·&nbsp;1.415&nbsp;<&nbsp;0.4748. However, if ''ρ''&nbsp;≥&nbsp;1.286''σ''<sup>3</sup>, then the estimate 
:<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le {0.3328 (\rho+0.429\sigma^3)\over \sigma^3\sqrt{n}},</math>
is even tighter.{{sfnp|Shevtsova|2011}}

{{harvtxt|Esseen|1956}} proved that the constant also satisfies the lower bound
: <math>
    C\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.40973 \approx \frac{1}{\sqrt{2\pi}} + 0.01079 .
  </math>