Editing Berry–Esseen theorem (section)

===Non-identically distributed summands===

:Let ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., be independent random variables with [[expected value|E]](''X''<sub>''i''</sub>) = 0, E(''X''<sub>''i''</sub><sup>2</sup>) = ''σ''<sub>''i''</sub><sup>2</sup> > 0, and E(|''X''<sub>''i''</sub>|<sup>3</sup>) = ''ρ''<sub>''i''</sub> < ∞. Also, let
::<math>S_n = {X_1 + X_2 + \cdots + X_n \over \sqrt{\sigma_1^2+\sigma_2^2+\cdots+\sigma_n^2} }</math>
:be the normalized ''n''-th partial sum. Denote ''F''<sub>''n''</sub> the [[cumulative distribution function|cdf]] of ''S''<sub>''n''</sub>, and Φ the cdf of the [[standard normal distribution]]. For the sake of convenience denote  
::<math>\vec{\sigma}=(\sigma_1,\ldots,\sigma_n),\ \vec{\rho}=(\rho_1,\ldots,\rho_n).</math>
:In 1941, [[Andrew C. Berry]] proved that for all ''n'' there exists an absolute constant ''C''<sub>1</sub> such that
::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_1\cdot\psi_1,\ \ \ \ (2)</math>
:where
::<math>\psi_1=\psi_1\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits_{i=1}^n\sigma_i^2}\Big)^{-1/2}\cdot\max_{1\le
i\le n}\frac{\rho_i}{\sigma_i^2}.</math>

:Independently, in 1942, [[Carl-Gustav Esseen]] proved that for all ''n'' there exists an absolute constant ''C''<sub>0</sub> such that
::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_0\cdot\psi_0, \ \ \ \ (3)</math>
:where
::<math>\psi_0=\psi_0\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits_{i=1}^n\sigma_i^2}\Big)^{-3/2}\cdot\sum\limits_{i=1}^n\rho_i.</math>

It is easy to make sure that ψ<sub>0</sub>≤ψ<sub>1</sub>. Due to this circumstance inequality (3) is conventionally called the Berry–Esseen inequality, and the quantity ψ<sub>0</sub> is called the Lyapunov fraction of the third order. Moreover, in the case where the summands ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> have identical distributions 
::<math>\psi_0=\psi_1=\frac{\rho_1}{\sigma_1^3\sqrt{n}},</math>
and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.

Regarding ''C''<sub>0</sub>, obviously, the lower bound established by {{harvtxt|Esseen|1956}} remains valid:
: <math>
    C_0\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} = 0.4097\ldots.
  </math>

The lower bound is exactly reached only for certain Bernoulli distributions (see {{harvtxt|Esseen|1956}} for their explicit expressions).

The upper bounds for ''C''<sub>0</sub> were subsequently lowered from Esseen's original estimate 7.59 to 0.5600.<ref>{{harvtxt|Esseen|1942}}; {{harvtxt|Zolotarev|1967}}; {{harvtxt|van Beek|1972}}; {{harvtxt|Shiganov|1986}}; {{harvtxt|Tyurin|2009}}; {{harvtxt|Tyurin|2010}}; {{harvtxt|Shevtsova|2010}}.</ref>