Editing Central limit theorem (section)

===Convex body===
{{math theorem | math_statement = There exists a sequence {{math|''ε<sub>n</sub>'' ↓ 0}} for which the following holds. Let {{math|''n'' ≥ 1}}, and let random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} have a [[Logarithmically concave function|log-concave]] [[Joint density function|joint density]] {{mvar|f}} such that {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = ''f''({{abs|''x''<sub>1</sub>}}, ..., {{abs|''x<sub>n</sub>''}})}} for all {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}},  and {{math|1=E(''X''{{su|b=''k''|p=2}}) = 1}} for all {{math|1=''k'' = 1, ..., ''n''}}. Then the distribution of

<math display="block"> \frac{X_1+\cdots+X_n}{\sqrt n} </math>

is {{mvar|ε<sub>n</sub>}}-close to <math display="inline"> \mathcal{N}(0, 1)</math> in the [[Total variation distance of probability measures|total variation distance]].{{sfnp|Klartag|2007|loc=Theorem 1.2}}}}

These two {{mvar|ε<sub>n</sub>}}-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = const · exp(−({{abs|''x''<sub>1</sub>}}<sup>''α''</sup> + ⋯ + {{abs|''x<sub>n</sub>''}}<sup>''α''</sup>)<sup>''β''</sup>)}} where {{math|''α'' > 1}} and {{math|''αβ'' > 1}}. If {{math|1=''β'' = 1}} then {{math|''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} factorizes into {{math|const · exp (−{{abs|''x''<sub>1</sub>}}<sup>''α''</sup>) … exp(−{{abs|''x<sub>n</sub>''}}<sup>''α''</sup>), }} which means {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} are independent. In general, however, they are dependent.

The condition {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = ''f''({{abs|''x''<sub>1</sub>}}, ..., {{abs|''x<sub>n</sub>''}})}} ensures that {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} are of zero mean and [[uncorrelated]];{{Citation needed|date=June 2012}} still, they need not be independent, nor even [[Pairwise independence|pairwise independent]].{{Citation needed|date=June 2012}} By the way, pairwise independence cannot replace independence in the classical central limit theorem.{{sfnp|Durrett|2004|loc=Section 2.4, Example 4.5}}

Here is a [[Berry–Esseen theorem|Berry–Esseen]] type result.

{{math theorem | math_statement = Let {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} satisfy the assumptions of the previous theorem, then{{Sfnp|Klartag|2008|loc=Theorem 1}}

<math display="block"> \left| \mathbb{P} \left( a \le \frac{ X_1+\cdots+X_n }{ \sqrt n } \le b \right) - \frac1{\sqrt{2\pi}} \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le \frac{C}{n} </math>

for all {{math|''a'' < ''b''}}; here {{mvar|C}} is a [[mathematical constant|universal (absolute) constant]]. Moreover, for every {{math|''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' ∈ '''R'''}} such that {{math|1=''c''{{su|b=1|p=2}} + ⋯ + ''c''{{su|b=''n''|p=2}} = 1}},

<math display="block"> \left| \mathbb{P} \left( a \le c_1 X_1+\cdots+c_n X_n \le b \right) - \frac{1}{\sqrt{2\pi}} \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le C \left( c_1^4+\dots+c_n^4 \right). </math>}}

The distribution of {{math|{{sfrac|''X''<sub>1</sub> + ⋯ + ''X<sub>n</sub>''|{{sqrt|''n''}}}}}} need not be approximately normal (in fact, it can be uniform).{{sfnp|Klartag|2007|loc=Theorem 1.1}} However, the distribution of {{math|''c''<sub>1</sub>''X''<sub>1</sub> + ⋯ + ''c<sub>n</sub>X<sub>n</sub>''}} is close to <math display="inline"> \mathcal{N}(0, 1)</math> (in the total variation distance) for most vectors {{math|(''c''<sub>1</sub>, ..., ''c<sub>n</sub>'')}} according to the uniform distribution on the sphere {{math|1=''c''{{su|b=1|p=2}} + ⋯ + ''c''{{su|b=''n''|p=2}} = 1}}.