Editing Central limit theorem (section)

===Multidimensional CLT===
Proofs that use characteristic functions can be extended to cases where each individual <math display="inline">\mathbf{X}_i</math> is a [[random vector]] in {{nowrap|<math display="inline">\R^k</math>,}} with mean vector <math display="inline">\boldsymbol\mu = \operatorname E[\mathbf{X}_i]</math> and [[covariance matrix]] <math display="inline">\mathbf{\Sigma}</math> (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a [[multivariate normal distribution]].<ref name="vanderVaart">{{Cite book |last=van der Vaart |first=A.W. |title=Asymptotic statistics |year=1998 |publisher=Cambridge University Press |location=New York, NY |isbn=978-0-521-49603-2 |lccn=98015176}}</ref> Summation of these vectors is done component-wise. 

For <math>i = 1, 2, 3, \ldots,</math> let

<math display="block">\mathbf{X}_i = \begin{bmatrix} X_{i}^{(1)} \\ \vdots \\ X_{i}^{(k)} \end{bmatrix}</math>

be independent random vectors. The sum of the random vectors <math>\mathbf{X}_1, \ldots, \mathbf{X}_n</math> is

<math display="block">\sum_{i=1}^{n} \mathbf{X}_i = \begin{bmatrix} X_{1}^{(1)} \\ \vdots \\ X_{1}^{(k)} 
\end{bmatrix} + \begin{bmatrix} X_{2}^{(1)} \\ \vdots \\ X_{2}^{(k)} \end{bmatrix} + \cdots + \begin{bmatrix} X_{n}^{(1)} \\ \vdots \\ X_{n}^{(k)} \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^{n} X_{i}^{(1)} \\ \vdots \\ \sum_{i=1}^{n} X_{i}^{(k)} \end{bmatrix}</math>

and their average is

<math display="block">\mathbf{\bar X_n} = \begin{bmatrix} \bar X_{i}^{(1)} \\ \vdots \\ \bar X_{i}^{(k)} \end{bmatrix} = \frac{1}{n} \sum_{i=1}^{n} \mathbf{X}_i.</math>

Therefore,

<math display="block">\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left[ \mathbf{X}_i - \operatorname E \left( \mathbf{X}_i \right) \right] = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} ( \mathbf{X}_i - \boldsymbol\mu ) = \sqrt{n}\left(\overline{\mathbf{X}}_n - \boldsymbol\mu\right). </math>

The multivariate central limit theorem states that

<math display="block">\sqrt{n}\left( \overline{\mathbf{X}}_n - \boldsymbol\mu \right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}_k(0,\boldsymbol\Sigma),</math>
where the [[covariance matrix]] <math>\boldsymbol{\Sigma}</math> is equal to
<math display="block"> \boldsymbol\Sigma = \begin{bmatrix}
{\operatorname{Var} \left (X_{1}^{(1)} \right)} & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(2)} \right) & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(k)} \right) \\
\operatorname{Cov} \left (X_{1}^{(2)},X_{1}^{(1)} \right) & \operatorname{Var} \left( X_{1}^{(2)} \right) & \operatorname{Cov} \left(X_{1}^{(2)},X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left(X_{1}^{(2)},X_{1}^{(k)} \right) \\
\operatorname{Cov}\left (X_{1}^{(3)},X_{1}^{(1)} \right) & \operatorname{Cov} \left (X_{1}^{(3)},X_{1}^{(2)} \right) & \operatorname{Var} \left (X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1}^{(3)},X_{1}^{(k)} \right) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(1)} \right) & \operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(2)} \right) & \operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(3)} \right) & \cdots & \operatorname{Var} \left (X_{1}^{(k)} \right) \\
\end{bmatrix}~.</math>

The multivariate central limit theorem can be proved using the [[Cramér–Wold theorem]].<ref name="vanderVaart"/>

The rate of convergence is given by the following [[Berry–Esseen theorem|Berry–Esseen]] type result:

{{math theorem | name = Theorem<ref>{{cite web |first=Ryan |last=O’Donnell | author-link = Ryan O'Donnell (computer scientist) |year=2014 |title=Theorem&nbsp;5.38 |url=http://www.contrib.andrew.cmu.edu/~ryanod/?p=866 |access-date=2017-10-18 |archive-date=2019-04-08 |archive-url=https://web.archive.org/web/20190408054104/http://www.contrib.andrew.cmu.edu/~ryanod/?p=866 |url-status=dead }}</ref> | math_statement =
Let <math>X_1, \dots, X_n, \dots</math> be independent <math>\R^d</math>-valued random vectors, each having mean zero. Write <math>S =\sum^n_{i=1}X_i</math> and assume <math>\Sigma = \operatorname{Cov}[S]</math> is invertible. Let <math>Z \sim \mathcal{N}(0,\Sigma)</math> be a <math>d</math>-dimensional Gaussian with the same mean and same covariance matrix as <math>S</math>. Then for all convex sets {{nowrap|<math>U \subseteq \R^d</math>,}}

<math display="block">\left|\mathbb{P}[S \in U] - \mathbb{P}[Z \in U]\right| \le C \, d^{1/4} \gamma~,</math>
where <math>C</math> is a universal constant, {{nowrap|<math>\gamma = \sum^n_{i=1} \operatorname E \left[\left\| \Sigma^{-1/2}X_i\right\|^3_2\right]</math>,}} and <math>\|\cdot\|_2</math> denotes the Euclidean norm on {{nowrap|<math>\R^d</math>.}}
}}

It is unknown whether the factor <math display="inline">d^{1/4}</math> is necessary.<ref>{{cite journal |first=V. |last=Bentkus |title=A Lyapunov-type bound in <math>\R^d</math> |journal=Theory Probab. Appl. |volume=49 |year=2005 |issue=2 |pages=311–323 |doi=10.1137/S0040585X97981123 }}</ref>