Editing Central limit theorem

{{Short description|Fundamental theorem in probability theory and statistics}}
{{use dmy dates|cs1-dates=ly|date=July 2023}}
{{Infobox mathematical statement
| name = Central Limit Theorem
| image = [[File:IllustrationCentralTheorem.png|300px|class=skin-invert-image]]
| field = [[Probability theory]]
| type = [[Theorem]]
| statement = The scaled sum of a sequence of [[Independent and identically distributed random variables|i.i.d. random variables]] with finite positive [[variance]] converges in distribution to the [[normal distribution]].
| generalizations = [[Lindeberg's condition | Lindeberg's CLT]]
}}

In [[probability theory]], the '''central limit theorem''' ('''CLT''') states that, under appropriate conditions, the [[Probability distribution|distribution]] of a normalized version of the sample mean converges to a [[Normal distribution#Standard normal distribution|standard normal distribution]]. This holds even if the original variables themselves are not [[Normal distribution|normally distributed]]. There are several versions of the CLT, each applying in the context of different conditions.

The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late as 1920.{{sfnp|Fischer|2011|p={{page needed|date=July 2023}}}}

In [[statistics]], the CLT can be stated as: let <math>X_1, X_2, \dots, X_n</math> denote a [[Sampling (statistics)|statistical sample]] of size <math>n</math> from a population with [[expected value]] (average) <math>\mu</math> and finite positive [[variance]] <math>\sigma^2</math>, and let <math>\bar{X}_{n}</math> denote the sample mean (which is itself a [[random variable]]). Then the [[Convergence of random variables#Convergence in distribution|limit as <math>n\to\infty</math> of the distribution]] of <math>(\bar{X}_n-\mu) \sqrt{n} </math> is a normal distribution with mean <math> 0 </math> and variance <math>\sigma^2</math>.<ref>{{Cite book |last1=Montgomery |first1=Douglas C. |title=Applied Statistics and Probability for Engineers |edition=6th |last2=Runger |first2=George C. |publisher=Wiley |year=2014 |isbn=9781118539712 |page=241}}</ref>

In other words, suppose that a large sample of [[Random variate|observations]] is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average ([[arithmetic mean]]) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the [[probability distribution]] of these averages will closely approximate a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be [[independent and identically distributed]] (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.

The earliest version of this theorem, that the normal distribution may be used as an approximation to the [[binomial distribution]], is the [[de Moivre–Laplace theorem]].

==Independent sequences==
[[File:IllustrationCentralTheorem.png|thumb|upright=1.4 |right|Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.<ref>{{cite book |last=Rouaud |first=Mathieu |title=Probability, Statistics and Estimation|year=2013 |page=10 |url=http://www.incertitudes.fr/book.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf |archive-date=2022-10-09 |url-status=live}}</ref>]]

===Classical CLT===
Let <math>\{X_1, \ldots, X_n}\</math> be a sequence of [[Independent and identically distributed random variables|i.i.d. random variables]] having a distribution with [[expected value]] given by <math>\mu</math> and finite [[variance]] given by <math>\sigma^2.</math> Suppose we are interested in the [[sample mean|sample average]]

<math display="block">\bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}.</math>

By the [[law of large numbers]], the sample average [[Almost sure convergence|converges almost surely]] (and therefore also [[Convergence in probability|converges in probability]]) to the expected value <math>\mu</math> as <math>n\to\infty.</math>

The classical central limit theorem describes the size and the distributional form of the {{linktext|stochastic}} fluctuations around the deterministic number <math>\mu</math> during this convergence. More precisely, it states that as <math>n</math> gets larger, the distribution of the normalized mean <math>\sqrt{n}(\bar{X}_n - \mu)</math>, i.e. the difference between the sample average <math>\bar{X}_n</math> and its limit <math>\mu,</math> scaled by the factor <math>\sqrt{n}</math>, approaches the [[normal distribution]] with mean <math>0</math> and variance <math>\sigma^2.</math> For large enough <math>n,</math> the distribution of <math>\bar{X}_n</math> gets arbitrarily close to the normal distribution with mean <math>\mu</math> and variance <math>\sigma^2/n.</math>

The usefulness of the theorem is that the distribution of <math>\sqrt{n}(\bar{X}_n - \mu)</math> approaches normality regardless of the shape of the distribution of the individual <math>X_i.</math> Formally, the theorem can be stated as follows:

{{math theorem | name = Lindeberg–Lévy CLT | math_statement =
Suppose <math> X_1, X_2, X_3 \ldots</math> is a sequence of [[independent and identically distributed|i.i.d.]] random variables with <math>\operatorname E[X_i] = \mu</math> and <math>\operatorname{Var}[X_i] = \sigma^2 < \infty.</math> Then, as <math>n</math> approaches infinity, the random variables <math>\sqrt{n}(\bar{X}_n - \mu)</math> [[convergence in distribution|converge in distribution]] to a [[normal distribution|normal]] <math>\mathcal{N}(0, \sigma^2)</math>:{{sfnp|Billingsley|1995|p=357}}

<math display="block">\sqrt{n}\left(\bar{X}_n - \mu\right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}\left(0,\sigma^2\right) .</math>}}

In the case <math>\sigma > 0,</math> convergence in distribution means that the [[cumulative distribution function]]s of <math>\sqrt{n}(\bar{X}_n - \mu)</math> converge pointwise to the cdf of the <math>\mathcal{N}(0, \sigma^2)</math> distribution: for every real number <math>z,</math>

<math display="block">\lim_{n\to\infty} \mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] = \lim_{n\to\infty} \mathbb{P}\left[\frac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma } \le \frac{z}{\sigma}\right]= 
\Phi\left(\frac{z}{\sigma}\right) ,</math>

where <math>\Phi(z)</math> is the standard normal cdf evaluated at <math>z.</math> The convergence is uniform in <math>z</math> in the sense that

<math display="block">\lim_{n\to\infty}\;\sup_{z\in\R}\;\left|\mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] - \Phi\left(\frac{z}{\sigma}\right)\right| = 0~,</math>

where <math>\sup</math> denotes the least upper bound (or [[supremum]]) of the set.{{sfnp|Bauer|2001|loc=Theorem 30.13|p=199}}

===Lyapunov CLT===
In this variant of the central limit theorem the random variables <math display="inline">X_i</math> have to be independent, but not necessarily identically distributed. The theorem also requires that random variables <math display="inline">\left| X_i\right|</math> have [[moment (mathematics)|moment]]s of some order {{nowrap|<math display="inline">(2+\delta)</math>,}} and that the rate of growth of these moments is limited by the Lyapunov condition given below.

{{math theorem | name = Lyapunov CLT{{sfnp|Billingsley|1995|p=362}} | math_statement =
Suppose <math display="inline">\{X_1, \ldots, X_n, \ldots\}</math> is a sequence of independent random variables, each with finite expected value <math display="inline">\mu_i</math> and variance {{nowrap|<math display="inline">\sigma_i^2</math>.}} Define

<math display="block">s_n^2 = \sum_{i=1}^n \sigma_i^2 .</math>

If for some {{nowrap|<math display="inline">\delta > 0</math>,}} ''Lyapunov’s condition''

<math display="block">\lim_{n\to\infty} \; \frac{1}{s_{n}^{2+\delta}} \, \sum_{i=1}^{n} \operatorname E\left[\left|X_{i} - \mu_{i}\right|^{2+\delta}\right] = 0</math>

is satisfied, then a sum of <math display="inline">\frac{X_i - \mu_i}{s_n}</math> converges in distribution to a standard normal random variable, as <math display="inline">n</math> goes to infinity:

<math display="block">\frac{1}{s_n}\,\sum_{i=1}^{n} \left(X_i - \mu_i\right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}(0,1) .</math>}}

In practice it is usually easiest to check Lyapunov's condition for {{nowrap|<math display="inline">\delta = 1</math>.}}

If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.

===Lindeberg (-Feller) CLT===
{{Main|Lindeberg's condition}}

In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from [[Jarl Waldemar Lindeberg|Lindeberg]] in 1920).

Suppose that for every <math display="inline">\varepsilon > 0</math>,

<math display="block"> \lim_{n \to \infty} \frac{1}{s_n^2}\sum_{i = 1}^{n} \operatorname E\left[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\left\{\left| X_i - \mu_i \right| > \varepsilon s_n \right\}} \right] = 0</math>

where <math display="inline">\mathbf{1}_{\{\ldots\}}</math> is the [[indicator function]]. Then the distribution of the standardized sums

<math display="block">\frac{1}{s_n}\sum_{i = 1}^n \left( X_i - \mu_i \right)</math>

converges towards the standard normal distribution {{nowrap|<math display="inline">\mathcal{N}(0, 1)</math>.}}

===CLT for the sum of a random number of random variables===

Rather than summing an integer number <math>n</math> of random variables and taking <math>n \to \infty</math>, the sum can be of a random number <math>N</math> of random variables, with conditions on <math>N</math>.

{{math theorem | name = Robbins CLT<ref>{{cite journal |last1=Robbins |first1=Herbert |title=The asymptotic distribution of the sum of a random number of random variables |journal=Bull. Amer. Math. Soc. |date=1948 |volume=54 |issue=12 |pages=1151–1161 |doi=10.1090/S0002-9904-1948-09142-X |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-12/The-asymptotic-distribution-of-the-sum-of-a-random-number/bams/1183513324.full|doi-access=free }}</ref><ref>{{cite book |last1=Chen |first1=Louis H.Y. |last2=Goldstein |first2=Larry |last3=Shao |first3=Qi-Man |title=Normal Approximation by Stein's Method |date=2011 |publisher=Springer-Verlag |location=Berlin Heidelberg |pages=270–271}}</ref> | math_statement =
Let <math>\{X_i, i \geq 1\}</math> be independent, identically distributed random variables with <math>E(X_i) = \mu</math> and <math>\text{Var}(X_i) = \sigma^2</math>, and let <math>\{N_n, n \geq 1\}</math> be a sequence of non-negative integer-valued random variables that are independent of <math>\{X_i, i \geq 1\}</math>. Assume for each <math>n = 1, 2, \dots</math> that <math>E(N_n^2) < \infty</math> and

<math display="block">
  \frac{N_n - E(N_n)}{\sqrt{\text{Var}(N_n)}} \xrightarrow{\quad d \quad} \mathcal{N}(0,1)
</math>

where <math>\xrightarrow{\,d\,}</math> denotes convergence in distribution and <math>\mathcal{N}(0,1)</math> is the normal distribution with mean 0, variance 1.
Then

<math display="block">
  \frac{\sum_{i=1}^{N_n} X_i - \mu E(N_n)}{\sqrt{\sigma^2E(N_n) + \mu^2\text{Var}(N_n)}} \xrightarrow{\quad d \quad} \mathcal{N}(0,1)
</math>
}}

===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>

== The generalized central limit theorem ==

The generalized central limit theorem (GCLT) was an effort of multiple mathematicians ([[Sergei Natanovich Bernstein|Bernstein]], [[Jarl Waldemar Lindeberg|Lindeberg]], [[Paul Lévy (mathematician)|Lévy]], [[William Feller|Feller]], [[Andrey Kolmogorov|Kolmogorov]], and others) over the period from 1920 to 1937.<ref>{{cite journal |last1=Le Cam |first1=L. |title=The Central Limit Theorem around 1935 |journal=Statistical Science |date=February 1986 |volume=1 |issue=1 |pages=78–91 |jstor=2245503}}</ref> The first published complete proof of the GCLT was in 1937 by [[Paul Lévy (mathematician)|Paul Lévy]] in French.<ref>{{cite book |last1=Lévy |first1=Paul |title=Theorie de l'addition des variables aleatoires |lang=fr |trans-title=Combination theory of unpredictable variables |date=1937 |publisher=Gauthier-Villars |location=Paris}}</ref> An English language version of the complete proof of the GCLT is available in the translation of [[Boris Vladimirovich Gnedenko|Gnedenko]] and [[Andrey Kolmogorov|Kolmogorov]]'s 1954 book.<ref>{{cite book |last1=Gnedenko |first1=Boris Vladimirovich |last2=Kologorov |first2=Andreĭ Nikolaevich |last3=Doob |first3=Joseph L. |last4=Hsu |first4=Pao-Lu |title=Limit distributions for sums of independent random variables |date=1968 |publisher=Addison-wesley |location=Reading, MA}}</ref>

The statement of the GCLT is as follows:<ref>{{cite book |last1=Nolan |first1=John P. |title=Univariate stable distributions, Models for Heavy Tailed Data |series=Springer Series in Operations Research and Financial Engineering |date=2020 |publisher=Springer |location=Switzerland |doi=10.1007/978-3-030-52915-4 |isbn=978-3-030-52914-7 |s2cid=226648987 |url=https://doi.org/10.1007/978-3-030-52915-4}}</ref>

:''A non-degenerate random variable'' ''Z'' ''is [[stable distribution|''α''-stable]] for some'' 0 < ''α'' ≤ 2 ''if and only if there is an independent, identically distributed sequence of random variables'' ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ''... and constants'' ''a''<sub>''n''</sub> > 0, ''b''<sub>''n''</sub> ∈ ℝ ''with''

::''a''<sub>''n''</sub> (''X''<sub>1</sub> + ... + ''X''<sub>''n''</sub>) − ''b''<sub>''n''</sub> → ''Z''.

:''Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy'' ''F''<sub>''n''</sub>(''y'') → ''F''(''y'') ''at all continuity points of'' ''F.''

In other words, if sums of independent, identically distributed random variables converge in distribution to some ''Z'', then ''Z'' must be a [[stable distribution]].

==Dependent processes==

===CLT under weak dependence===
A useful generalization of a sequence of independent, identically distributed random variables is a [[Mixing (mathematics)|mixing]] random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially [[Mixing (mathematics)#Mixing in stochastic processes|strong mixing]] (also called α-mixing) defined by <math display="inline">\alpha(n) \to 0</math> where <math display="inline">\alpha(n)</math> is so-called [[Mixing (mathematics)#Mixing in stochastic processes|strong mixing coefficient]].

A simplified formulation of the central limit theorem under strong mixing is:{{sfnp|Billingsley|1995|loc=Theorem 27.4}}

{{math theorem | math_statement = Suppose that <math display="inline">\{X_1, \ldots, X_n, \ldots\}</math> is stationary and <math>\alpha</math>-mixing with <math display="inline">\alpha_n = O\left(n^{-5}\right) </math> and that <math display="inline">\operatorname E[X_n] = 0</math> and {{nowrap|<math display="inline">\operatorname E[X_n^{12}] < \infty</math>.}} Denote {{nowrap|<math display="inline">S_n = X_1 + \cdots + X_n</math>,}} then the limit

<math display="block"> \sigma^2 = \lim_{n\rightarrow\infty} \frac{\operatorname E\left(S_n^2\right)}{n} </math>

exists, and if <math display="inline">\sigma \ne 0</math> then <math display="inline">\frac{S_n}{\sigma\sqrt{n}}</math> converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math>.}}

In fact,

<math display="block">\sigma^2  = \operatorname E\left(X_1^2\right) + 2 \sum_{k=1}^{\infty} \operatorname E\left(X_1 X_{1+k}\right),</math>

where the series converges absolutely.

The assumption <math display="inline">\sigma \ne 0</math> cannot be omitted, since the asymptotic normality fails for <math display="inline">X_n = Y_n - Y_{n-1}</math> where <math display="inline">Y_n</math> are another [[stationary sequence]].

There is a stronger version of the theorem:{{sfnp|Durrett|2004|loc=Sect. 7.7(c), Theorem 7.8}} the assumption <math display="inline">\operatorname E\left[X_n^{12}\right] < \infty</math> is replaced with {{nowrap|<math display="inline">\operatorname E\left[{\left|X_n\right|}^{2+\delta}\right] < \infty</math>,}} and the assumption <math display="inline">\alpha_n = O\left(n^{-5}\right) </math> is replaced with

<math display="block">\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.</math>

Existence of such <math display="inline">\delta > 0</math> ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see {{harv|Bradley|2007}}.

===Martingale difference CLT===
{{Main|Martingale central limit theorem}}
{{math theorem | math_statement = Let a [[Martingale (probability theory)|martingale]] <math display="inline">M_n</math> satisfy
* <math> \frac1n \sum_{k=1}^n \operatorname E\left[\left(M_k-M_{k-1}\right)^2 \mid M_1,\dots,M_{k-1}\right] \to 1 </math>  in probability as {{math|''n'' → ∞}},
* for every {{math|''ε'' > 0}}, <math> \frac1n \sum_{k=1}^n{\operatorname E\left[\left(M_k-M_{k-1} \right)^2\mathbf{1}\left[|M_k-M_{k-1}|>\varepsilon\sqrt{n}\right]\right]} \to 0 </math>  as {{math|''n'' → ∞}},

then <math display="inline">\frac{M_n}{\sqrt{n}}</math> converges in distribution to <math display="inline">\mathcal{N}(0, 1)</math> as <math display="inline">n \to \infty</math>.{{sfnp|Durrett|2004|loc=Sect. 7.7, Theorem 7.4}}{{sfnp|Billingsley|1995 |loc=Theorem 35.12}}}}

==Remarks==

===Proof of classical CLT===
The central limit theorem has a proof using [[characteristic function (probability theory)|characteristic functions]].<ref>{{cite book|url=https://jhupbooks.press.jhu.edu/content/introduction-stochastic-processes-physics|title=An Introduction to Stochastic Processes in Physics|publisher=Johns Hopkins University Press|year=2003 |doi=10.56021/9780801868665 |access-date=2016-08-11 |last1=Lemons |first1=Don |isbn=9780801876387 }}</ref> It is similar to the proof of the (weak) [[Proof of the law of large numbers|law of large numbers]].

Assume <math display="inline">\{X_1, \ldots, X_n, \ldots \}</math> are independent and identically distributed random variables, each with mean <math display="inline">\mu</math> and finite variance {{nowrap|<math display="inline">\sigma^2</math>.}} The sum <math display="inline">X_1 + \cdots + X_n</math> has [[Linearity of expectation|mean]] <math display="inline">n\mu</math> and [[Variance#Sum of uncorrelated variables (Bienaymé formula)|variance]] {{nowrap|<math display="inline">n\sigma^2</math>.}} Consider the random variable

<math display="block">Z_n = \frac{X_1+\cdots+X_n - n \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{X_i - \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{1}{\sqrt{n}} Y_i,</math>

where in the last step we defined the new random variables {{nowrap|<math display="inline">Y_i = \frac{X_i - \mu}{\sigma} </math>,}} each with zero mean and unit variance {{nowrap|(<math display="inline">\operatorname{var}(Y) = 1</math>).}} The [[Characteristic function (probability theory)|characteristic function]] of <math display="inline">Z_n</math> is given by

<math display="block">\varphi_{Z_n}\!(t) = \varphi_{\sum_{i=1}^n {\frac{1}{\sqrt{n}}Y_i}}\!(t) \ =\ \varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right) \varphi_{Y_2}\!\! \left(\frac{t}{\sqrt{n}}\right)\cdots \varphi_{Y_n}\!\! \left(\frac{t}{\sqrt{n}}\right) \ =\ \left[\varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right)\right]^n,
</math>

where in the last step we used the fact that all of the <math display="inline">Y_i</math> are identically distributed. The characteristic function of <math display="inline">Y_1</math> is, by [[Taylor's theorem]],
<math display="block">\varphi_{Y_1}\!\left(\frac{t}{\sqrt{n}}\right) = 1 - \frac{t^2}{2n} + o\!\left(\frac{t^2}{n}\right), \quad \left(\frac{t}{\sqrt{n}}\right) \to 0</math>

where <math display="inline">o(t^2 / n)</math> is "[[Little-o notation|little {{mvar|o}} notation]]" for some function of <math display="inline">t</math> that goes to zero more rapidly than {{nowrap|<math display="inline">t^2 / n</math>.}} By the limit of the [[exponential function]] {{nowrap|(<math display="inline">e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n</math>),}} the characteristic function of <math>Z_n</math> equals

<math display="block">\varphi_{Z_n}(t) = \left(1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right) \right)^n \rightarrow e^{-\frac{1}{2} t^2}, \quad n \to \infty.</math>

All of the higher order terms vanish in the limit {{nowrap|<math display="inline">n\to\infty</math>.}} The right hand side equals the characteristic function of a standard normal distribution <math display="inline">\mathcal{N}(0, 1)</math>, which implies through [[Lévy continuity theorem|Lévy's continuity theorem]] that the distribution of <math display="inline">Z_n</math> will approach <math display="inline">\mathcal{N}(0,1)</math> as {{nowrap|<math display="inline">n\to\infty</math>.}} Therefore, the [[sample mean|sample average]]

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

is such that 

<math display="block">\frac{\sqrt{n}}{\sigma}(\bar{X}_n - \mu) = Z_n</math>

converges to the normal distribution {{nowrap|<math display="inline">\mathcal{N}(0, 1)</math>,}} from which the central limit theorem follows.

===Convergence to the limit===
The central limit theorem gives only an [[asymptotic distribution]]. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.{{citation needed|reason=Not immediately obvious, I didn't find a source via google|date=July 2016}}

The convergence in the central limit theorem is [[uniform convergence|uniform]] because the limiting cumulative distribution function is continuous. If the third central [[Moment (mathematics)|moment]] <math display="inline">\operatorname{E}\left[(X_1 - \mu)^3\right]</math> exists and is finite, then the speed of convergence is at least on the order of <math display="inline">1 / \sqrt{n}</math> (see [[Berry–Esseen theorem]]). [[Stein's method]]<ref name="stein1972">{{Cite journal| last = Stein |first=C. |author-link=Charles Stein (statistician)| title = A bound for the error in the normal approximation to the distribution of a sum of dependent random variables| journal = Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability| pages= 583–602| year = 1972|volume=6 |issue=2 | mr=402873 | zbl = 0278.60026| url=http://projecteuclid.org/euclid.bsmsp/1200514239 }}</ref> can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.<ref>{{Cite book|  title = Normal approximation by Stein's method|  publisher = Springer| year = 2011|last1=Chen |first1=L. H. Y. |last2=Goldstein |first2=L. |last3=Shao |first3=Q. M. |isbn = 978-3-642-15006-7}}</ref>

The convergence to the normal distribution is monotonic, in the sense that the [[information entropy|entropy]] of <math display="inline">Z_n</math> increases [[monotonic function|monotonically]] to that of the normal distribution.<ref name=ABBN/>

The central limit theorem applies in particular to sums of independent and identically distributed [[discrete random variable]]s.  A sum of [[discrete random variable]]s is still a [[discrete random variable]], so that we are confronted with a sequence of [[discrete random variable]]s whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the [[normal distribution]]).  This means that if we build a [[histogram]] of the realizations of the sum of {{mvar|n}} independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as {{mvar|n}} approaches infinity; this relation is known as [[de Moivre–Laplace theorem]]. The [[binomial distribution]] article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.

===Common misconceptions===
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.<ref>{{cite journal |last=Brewer |first=J. K. |date=1985 |title=Behavioral statistics textbooks: Source of myths and misconceptions? |journal=Journal of Educational Statistics |volume=10 |issue=3 |pages=252–268|doi=10.3102/10769986010003252 |s2cid=119611584 }}</ref><ref>Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, ''American Educational Research Association'' lecture 19 April 1995</ref><ref>{{cite journal |last1=Sotos |first1=A. E. C. |last2=Vanhoof |first2=S. |last3=Van den Noortgate |first3=W. |last4=Onghena |first4=P. |date=2007 |title=Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education |journal=Educational Research Review |volume=2 |issue=2 |pages=98–113|doi=10.1016/j.edurev.2007.04.001 |url=https://lirias.kuleuven.be/handle/123456789/136347 }}</ref> These include: 
* The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of [[iid]] random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables. 
* The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the [[Glivenko–Cantelli theorem]]. 
* The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,<ref>{{Cite web |date=2023-06-02 |title=Sampling distribution of the sample mean |format=video |website=Khan Academy |url=https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean |access-date=2023-10-08 |archive-url=https://web.archive.org/web/20230602200310/https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean |archive-date=2 June 2023 }}</ref> allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See [[Z-test]] for where the approximation holds.

===Relation to the law of large numbers===
The [[law of large numbers]] as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of {{math|S<sub>{{mvar|n}}</sub>}} as {{mvar|n}} approaches infinity?" In mathematical analysis, [[asymptotic series]] are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of <math display="inline">f(n)</math>:

<math display="block">f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O\big(\varphi_{3}(n)\big) \qquad  (n \to \infty).</math>

Dividing both parts by {{math|''φ''<sub>1</sub>(''n'')}} and taking the limit will produce {{math|''a''<sub>1</sub>}}, the coefficient of the highest-order term in the expansion, which represents the rate at which {{math|''f''(''n'')}} changes in its leading term.

<math display="block">\lim_{n\to\infty} \frac{f(n)}{\varphi_{1}(n)} = a_1.</math>

Informally, one can say: "{{math|''f''(''n'')}} grows approximately as {{math|''a''<sub>1</sub>''φ''<sub>1</sub>(''n'')}}". Taking the difference between {{math|''f''(''n'')}} and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about {{math|''f''(''n'')}}:

<math display="block">\lim_{n\to\infty} \frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)} = a_2 .</math>

Here one can say that the difference between the function and its approximation grows approximately as {{math|''a''<sub>2</sub>''φ''<sub>2</sub>(''n'')}}.  The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.

Informally, something along these lines happens when the sum, {{mvar|S<sub>n</sub>}}, of independent identically distributed random variables, {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, is studied in classical probability theory.{{Citation needed|date=April 2012}}  If each {{mvar|X<sub>i</sub>}} has finite mean {{mvar|μ}}, then by the law of large numbers, {{math|{{sfrac|''S<sub>n</sub>''|''n''}} → ''μ''}}.<ref>{{cite book|last=Rosenthal |first=Jeffrey Seth |date=2000 |title=A First Look at Rigorous Probability Theory |publisher=World Scientific |isbn=981-02-4322-7 |at=Theorem 5.3.4, p. 47}}</ref>  If in addition each {{mvar|X<sub>i</sub>}} has finite variance {{math|''σ''<sup>2</sup>}}, then by the central limit theorem,

<math display="block"> \frac{S_n-n\mu}{\sqrt{n}} \to \xi ,</math>

where {{mvar|ξ}} is distributed as {{math|''N''(0,''σ''<sup>2</sup>)}}.  This provides values of the first two constants in the informal expansion

<math display="block">S_n \approx \mu n+\xi \sqrt{n}. </math>

In the case where the {{mvar|X<sub>i</sub>}} do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:

<math display="block">\frac{S_n-a_n}{b_n} \rightarrow \Xi,</math>

or informally

<math display="block">S_n \approx a_n+\Xi b_n. </math>

Distributions {{math|Ξ}} which can arise in this way are called ''[[stable distribution|stable]]''.<ref>{{cite book|last=Johnson |first=Oliver Thomas |date=2004 |title=Information Theory and the Central Limit Theorem |publisher=Imperial College Press |isbn= 1-86094-473-6 |page= 88}}</ref>  Clearly, the normal distribution is stable, but there are also other stable distributions, such as the [[Cauchy distribution]], for which the mean or variance are not defined.  The scaling factor {{mvar|b<sub>n</sub>}} may be proportional to {{mvar|n<sup>c</sup>}}, for any {{math|''c'' ≥ {{sfrac|1|2}}}}; it may also be multiplied by a [[slowly varying function]] of {{mvar|n}}.<ref name=Uchaikin>{{cite book |first1=Vladimir V. |last1=Uchaikin |first2=V.M. |last2=Zolotarev |year=1999 |title=Chance and Stability: Stable distributions and their applications |publisher=VSP |isbn=90-6764-301-7 |pages=61–62}}</ref><ref>{{cite book|last1=Borodin |first1=A. N. |last2=Ibragimov |first2=I. A. |last3=Sudakov |first3=V. N. |date=1995 |title=Limit Theorems for Functionals of Random Walks |publisher=AMS Bookstore |isbn= 0-8218-0438-3 |at=Theorem 1.1, p. 8}}</ref>

The [[law of the iterated logarithm]] specifies what is happening "in between" the [[law of large numbers]] and the central limit theorem. Specifically it says that the normalizing function {{math|{{sqrt|''n'' log log ''n''}}}}, intermediate in size between {{mvar|n}} of the law of large numbers and {{math|{{sqrt|''n''}}}} of the central limit theorem, provides a non-trivial limiting behavior.

===Alternative statements of the theorem===

====Density functions====
The [[probability density function|density]] of the sum of two or more independent variables is the [[convolution]] of their densities (if these densities exist).  Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov<ref>{{Cite book|last=Petrov|first=V. V. |title=Sums of Independent Random Variables|year=1976|publisher=Springer-Verlag|location=New York-Heidelberg | isbn=9783642658099 | at=ch. 7|url=https://books.google.com/books?id=zSDqCAAAQBAJ}}</ref> for a particular local limit theorem for sums of [[independent and identically distributed random variables]].

====Characteristic functions====
Since the [[characteristic function (probability theory)|characteristic function]] of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.

An equivalent statement can be made about [[Fourier transform]]s, since the characteristic function is essentially a Fourier transform.

===Calculating the variance===
Let {{mvar|S<sub>n</sub>}} be the sum of {{mvar|n}} random variables. Many central limit theorems provide conditions such that {{math|{{mvar|S<sub>n</sub>}}/{{sqrt|Var({{mvar|S<sub>n</sub>}})}}}} converges in distribution to {{math|''N''(0,1)}} (the normal distribution with mean 0, variance 1) as {{math|{{mvar|n}} → ∞}}. In some cases, it is possible to find a constant {{math|''σ''<sup>2</sup>}} and function {{mvar|f(n)}} such that {{math|{{mvar|S<sub>n</sub>}}/(σ{{sqrt|{{mvar|n⋅f}}({{mvar|n}})}})}} converges in distribution to {{math|''N''(0,1)}} as {{math|{{mvar|n}}→ ∞}}.

{{math theorem | name = Lemma<ref>{{cite journal|last1=Hew|first1=Patrick Chisan|title=Asymptotic distribution of rewards accumulated by alternating renewal processes|journal=Statistics and Probability Letters|date=2017|volume=129 |pages=355–359 |doi=10.1016/j.spl.2017.06.027}}</ref> | math_statement = Suppose <math>X_1, X_2, \dots</math> is a sequence of real-valued and strictly stationary random variables with <math>\operatorname E(X_i) = 0</math> for all {{nowrap|<math>i</math>,}} {{nowrap|<math>g : [0,1] \to \R</math>,}} and {{nowrap|<math>S_n = \sum_{i=1}^{n} g\left(\tfrac{i}{n}\right) X_i</math>.}} Construct

<math display="block">\sigma^2 = \operatorname E(X_1^2) + 2\sum_{i=1}^\infty \operatorname E(X_1 X_{1+i})</math>

# If <math>\sum_{i=1}^\infty \operatorname E(X_1 X_{1+i})</math> is absolutely convergent, <math>\left| \int_0^1 g(x)g'(x) \, dx\right| < \infty</math>, and <math>0 < \int_0^1 (g(x))^2 dx < \infty</math> then <math>\mathrm{Var}(S_n)/(n \gamma_n) \to \sigma^2</math> as <math>n \to \infty</math> where {{nowrap|<math>\gamma_n = \frac{1}{n}\sum_{i=1}^{n} \left(g\left(\tfrac{i}{n}\right)\right)^2</math>.}}
# If in addition <math>\sigma > 0</math> and <math>S_n/\sqrt{\mathrm{Var}(S_n)}</math> converges in distribution to <math>\mathcal{N}(0,1)</math> as <math>n \to \infty</math> then <math>S_n/(\sigma\sqrt{n \gamma_n})</math> also converges in distribution to <math>\mathcal{N}(0,1)</math> as {{nowrap|<math>n \to \infty</math>.}}
}}

==Extensions==

===Products of positive random variables===
The [[logarithm]] of a product is simply the sum of the logarithms of the factors.  Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a [[log-normal distribution]].  Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different [[random]] factors, so they follow a log-normal distribution.  This multiplicative version of the central limit theorem is sometimes called [[Gibrat's law]].

Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.<ref name=Rempala/>

==Beyond the classical framework==
Asymptotic normality, that is, [[Convergence in distribution|convergence]] to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.

===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}}.

===Lacunary trigonometric series===
{{math theorem | name = Theorem ([[Raphaël Salem|Salem]]–[[Antoni Zygmund|Zygmund]]) | math_statement =
Let {{mvar|U}} be a random variable distributed uniformly on {{math|(0,2π)}}, and {{math|1=''X<sub>k</sub>'' = ''r<sub>k</sub>'' cos(''n<sub>k</sub>U'' + ''a<sub>k</sub>'')}}, where
* {{mvar|n<sub>k</sub>}} satisfy the lacunarity condition: there exists {{math|''q'' > 1}} such that {{math|''n''<sub>''k'' + 1</sub> ≥ ''qn''<sub>''k''</sub>}} for all {{mvar|k}},
* {{mvar|r<sub>k</sub>}} are such that<br /><math display="block"> r_1^2 + r_2^2 + \cdots = \infty \quad\text{ and }\quad \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0, </math>
* {{math|0 ≤ ''a''<sub>''k''</sub> < 2π}}.
Then<ref name=Zygmund/>{{sfnp|Gaposhkin|1966|loc=Theorem 2.1.13}}

<math display="block"> \frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} } </math>

converges in distribution to <math display="inline"> \mathcal{N}\big(0, \frac{1}{2}\big)</math>.}}

===Gaussian polytopes===
{{math theorem | math_statement =
Let {{math|''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub>}} be independent random points on the plane {{math|'''R'''<sup>2</sup>}} each having the two-dimensional standard normal distribution. Let {{mvar|K<sub>n</sub>}} be the [[convex hull]] of these points, and {{mvar|X<sub>n</sub>}} the area of {{mvar|K<sub>n</sub>}} Then{{sfnp|Bárány|Vu|2007|loc=Theorem 1.1}}

<math display="block"> \frac{ X_n - \operatorname E (X_n) }{ \sqrt{\operatorname{Var} (X_n)} } </math>
converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math> as {{mvar|n}} tends to infinity.}}

The same also holds in all dimensions greater than 2.

The [[convex polytope|polytope]] {{mvar|K<sub>n</sub>}} is called a Gaussian random polytope.

A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.{{sfnp|Bárány|Vu|2007|loc=Theorem 1.2}}

===Linear functions of orthogonal matrices===
A linear function of a matrix {{math|'''M'''}} is a linear combination of its elements (with given coefficients), {{math|'''M''' ↦ tr('''AM''')}} where {{math|'''A'''}} is the matrix of the coefficients; see [[Trace (linear algebra)#Inner product]].

A random [[orthogonal matrix]] is said to be distributed uniformly, if its distribution is the normalized [[Haar measure]] on the [[orthogonal group]] {{math|O(''n'','''R''')}}; see [[Rotation matrix#Uniform random rotation matrices]].

{{math theorem | math_statement = Let {{math|'''M'''}} be a random orthogonal {{math|''n'' × ''n''}} matrix distributed uniformly, and {{math|'''A'''}} a fixed {{math|''n'' × ''n''}} matrix such that {{math|1=tr('''AA'''*) = ''n''}}, and let {{math|1=''X'' = tr('''AM''')}}. Then<ref name=Meckes/> the distribution of {{mvar|X}} is close to <math display="inline"> \mathcal{N}(0, 1)</math> in the total variation metric up to{{clarify|reason=what does up to mean here|date=June 2012}} {{math|{{sfrac|2{{sqrt|3}}|''n'' − 1}}}}.}}

===Subsequences===
{{math theorem | math_statement = Let random variables {{math|''X''<sub>1</sub>, ''X''<sub>2</sub>, ... ∈ ''L''<sub>2</sub>(Ω)}} be such that {{math|''X<sub>n</sub>'' → 0}} [[Weak convergence (Hilbert space)|weakly]] in {{math|''L''<sub>2</sub>(Ω)}} and {{math|''X''{{su|b=''n''|2}} → 1}} weakly in {{math|''L''<sub>1</sub>(Ω)}}. Then there exist integers {{math|''n''<sub>1</sub> < ''n''<sub>2</sub> < ⋯}} such that

<math display="block"> \frac{ X_{n_1}+\cdots+X_{n_k} }{ \sqrt k }</math>

converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math> as {{mvar|k}} tends to infinity.{{sfnp|Gaposhkin|1966|loc=Sect. 1.5}}}}

===Random walk on a crystal lattice===

The central limit theorem may be established for the simple [[random walk]] on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.<ref>{{cite book |last1=Kotani |first1=M. |last2=Sunada |first2=Toshikazu |author-link2=Toshikazu Sunada |date=2003 |title=Spectral geometry of crystal lattices |publisher=Contemporary Math |volume=338 |pages=271–305 |isbn=978-0-8218-4269-0}}</ref><ref>{{cite book |author-link=Toshikazu Sunada |last=Sunada |first=Toshikazu |date=2012 |title=Topological Crystallography – With a View Towards Discrete Geometric Analysis|series=Surveys and Tutorials in the Applied Mathematical Sciences |volume=6 |publisher=Springer |isbn=978-4-431-54177-6}}</ref>

==Applications and examples==
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample [[statistic]]s to the normal distribution in controlled experiments.

{{multiple image |total_width=830 |align=center
|image1=Dice sum central limit theorem.svg |caption1=Comparison of probability density functions {{math|''p''(''k'')}} for the sum of {{mvar|n}} fair 6-sided dice to show their convergence to a normal distribution with increasing {{mvar|n}}, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
|image2=Empirical CLT - Figure - 040711.jpg |caption2=This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the [[Pearson's chi-squared test|chi-squared]] values that quantify the goodness of the fit (the fit is good if the reduced [[Pearson's chi-squared test|chi-squared]] value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/{{math|{{sqrt|''n''}}}}), which is called the standard deviation of the mean (since it refers to the spread of sample means).
}}

[[File:Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg|center|thumb|820px|Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.]]

===Regression===
[[Regression analysis]], and in particular [[ordinary least squares]], specifies that a [[dependent variable]] depends according to some function upon one or more [[independent variable]]s, with an additive [[Errors and residuals in statistics|error term]]. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.

===Other illustrations===
{{Main|Illustration of the central limit theorem}}
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.<ref name="Marasinghe">{{cite conference |last1=Marasinghe |first1=M. |last2=Meeker |first2=W. |last3=Cook |first3=D. |last4=Shin |first4=T. S. |date=Aug 1994 |title=Using graphics and simulation to teach statistical concepts |conference=Annual meeting of the American Statistician Association, Toronto, Canada}}</ref>

==History==
Dutch mathematician [[Henk Tijms]] writes:<ref name=Tijms/>

{{blockquote|The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician [[Abraham de Moivre]] who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician [[Pierre-Simon Laplace]] rescued it from obscurity in his monumental work ''Théorie analytique des probabilités'', which was published in 1812.  Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician [[Aleksandr Lyapunov]] defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.}}

Sir [[Francis Galton]] described the Central Limit Theorem in this way:<ref>{{cite book|last=Galton|first= F. |date=1889 |title=Natural Inheritance |url=http://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm |page= 66}}</ref>

{{blockquote|I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.}}

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by [[George Pólya]] in 1920 in the title of a paper.<ref name=Polya1920>{{Cite journal|last=Pólya|first=George|author-link=George Pólya|year=1920|title=Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem|trans-title=On the central limit theorem of probability calculation and the problem of moments |journal=[[Mathematische Zeitschrift]]|volume=8|pages=171–181 |language=de |url=http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0008|doi=10.1007/BF01206525 |issue=3–4|s2cid=123063388}}</ref><ref name=LC1986/> Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word ''central'' in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".<ref name=LC1986/> The abstract of the paper ''On the central limit theorem of calculus of probability and the problem of moments'' by Pólya<ref name=Polya1920/> in 1920 translates as follows.

{{blockquote|text=The occurrence of the Gaussian probability density {{math|1 {{=}} ''e''<sup>−''x''<sup>2</sup></sup>}} in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by [[Aleksandr Lyapunov|Liapounoff]]. ... }}

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as [[Augustin-Louis Cauchy|Cauchy]]'s, [[Friedrich Bessel|Bessel]]'s and [[Siméon Denis Poisson|Poisson]]'s contributions, is provided by Hald.<ref name=Hald/> Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by [[Richard von Mises|von Mises]], [[George Pólya|Pólya]], [[Jarl Waldemar Lindeberg|Lindeberg]], [[Paul Lévy (mathematician)|Lévy]], and [[Harald Cramér|Cramér]] during the 1920s, are given by Hans Fischer.{{sfnp|Fischer|2011|loc=Chapter 2; Chapter 5.2}} Le Cam describes a period  around 1935.<ref name=LC1986/> Bernstein<ref name=Bernstein/> presents a historical discussion focusing on the work of [[Pafnuty Chebyshev]] and his students [[Andrey Markov]] and [[Aleksandr Lyapunov]] that led to the first proofs of the CLT in a general setting.

A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of [[Alan Turing]]'s 1934 Fellowship Dissertation for [[King's College, Cambridge|King's College]] at the [[University of Cambridge]].  Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.<ref>{{cite journal |first=S. L. |last=Zabell |title=Alan Turing and the Central Limit Theorem |journal=American Mathematical Monthly |volume=102 |year=1995 |issue=6 |pages=483–494 |doi=10.1080/00029890.1995.12004608 }}</ref>

==See also==
* [[Asymptotic equipartition property]]
* [[Asymptotic distribution]]
* [[Bates distribution]]
* [[Benford's law]] – result of extension of CLT to product of random variables.
* [[Berry–Esseen theorem]]
* [[Central limit theorem for directional statistics]] – Central limit theorem applied to the case of directional statistics
* [[Delta method]] – to compute the limit distribution of a function of a random variable.
* [[Erdős–Kac theorem]] – connects the number of prime factors of an integer with the normal probability distribution
* [[Fisher–Tippett–Gnedenko theorem]] – limit theorem for extremum values (such as {{math|max{''X<sub>n</sub>''}<nowiki/>}})
* [[Irwin–Hall distribution]]
* [[Markov chain central limit theorem]]
* [[Normal distribution]]
* [[Tweedie distribution|Tweedie convergence theorem]] – a theorem that can be considered to bridge between the central limit theorem and the [[Poisson convergence theorem]]<ref name="Jørgensen-1997">{{cite book| last= Jørgensen|first= Bent | year = 1997| title = The Theory of Dispersion Models| publisher = Chapman & Hall | isbn = 978-0412997112}}</ref>
* [[Donsker's theorem]]

==Notes==
{{Reflist|30em|refs=

<ref name=ABBN>{{cite journal| first1=S.| last1=Artstein |author1-link=Shiri Artstein| first2=K. |last2=Ball |author2-link=Keith Martin Ball |first3=F. |last3=Barthe |author3-link=Franck Barthe |first4=A. |last4=Naor |author4-link=Assaf Naor |year=2004 |title=Solution of Shannon's Problem on the Monotonicity of Entropy |journal=Journal of the American Mathematical Society  |volume=17 |pages=975–982| doi=10.1090/S0894-0347-04-00459-X| issue=4 |doi-access=free}}</ref>

<ref name=Bernstein>{{cite book|author-link=Sergei Natanovich Bernstein|last=Bernstein|first=S. N. |date=1945 |contribution=On the work of P. L. Chebyshev in Probability Theory |title=Nauchnoe Nasledie P. L. Chebysheva. Vypusk Pervyi: Matematika |language=ru |trans-title=The Scientific Legacy of P. L. Chebyshev. Part I: Mathematics |editor-first=S. N.|editor-last= Bernstein. |publisher=Academiya Nauk SSSR |location=Moscow & Leningrad |page=174}}</ref>

<ref name=LC1986>{{cite journal| author-link=Lucien Le Cam|last=Le Cam |first= Lucien|year=1986 |url=http://projecteuclid.org/euclid.ss/1177013818 |title=The central limit theorem around 1935 |journal=Statistical Science |volume=1|issue=1|pages=78–91|doi=10.1214/ss/1177013818|doi-access=free}}</ref>

<ref name=Hald>{{cite book|last=Hald |first=Andreas |url=http://www.gbv.de/dms/goettingen/229762905.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.gbv.de/dms/goettingen/229762905.pdf |archive-date=2022-10-09 |url-status=live |title=A History of Mathematical Statistics from 1750 to 1930|isbn=978-0471179122 |date=22 April 1998 |publisher=Wiley |at=chapter 17}}</ref>

<ref name=Meckes>{{Cite journal|last=Meckes|first= Elizabeth|author-link=Elizabeth Meckes|year=2008|title=Linear functions on the classical matrix groups|journal=Transactions of the American Mathematical Society|volume=360|pages=5355–5366|doi=10.1090/S0002-9947-08-04444-9|issue=10 |arxiv=math/0509441 |s2cid= 11981408}}</ref>

<ref name=Rempala>{{cite journal |last1=Rempala |first1=G. |last2=Wesolowski |first2=J. |year=2002 |title=Asymptotics of products of sums and ''U''-statistics |url=https://projecteuclid.org/journals/electronic-communications-in-probability/volume-7/issue-none/Asymptotics-for-Products-of-Sums-and-U-statistics/10.1214/ECP.v7-1046.pdf |journal=Electronic Communications in Probability |volume=7 |pages=47–54 |doi=10.1214/ecp.v7-1046| doi-access=free}}</ref>

<ref name=Tijms>{{Cite book |first=Tijms |last=Henk |year=2004 |title=Understanding Probability: Chance Rules in Everyday Life |location=Cambridge |publisher=Cambridge University Press |isbn=0-521-54036-4 |page=169}}</ref>

<ref name=Zygmund>{{cite book|last=Zygmund|first=Antoni|author-link=Antoni Zygmund|orig-year=1959|title=Trigonometric Series|title-link= Trigonometric Series |publisher=Cambridge University Press|date=2003 |isbn= 0-521-89053-5 |at=vol. II, sect. XVI.5, Theorem 5-5}}</ref>
}}

==References==
*{{cite journal|last1=Bárány|first1=Imre|author-link1=Imre Bárány|last2=Vu|first2=Van|year=2007|title=Central limit theorems for Gaussian polytopes|journal=Annals of Probability|publisher=Institute of Mathematical Statistics |volume=35|issue=4|pages=1593–1621 |arxiv=math/0610192 |doi=10.1214/009117906000000791 |s2cid=9128253}}
*{{cite book|last=Bauer|first=Heinz|title=Measure and Integration Theory|publisher=de Gruyter |location=Berlin |year=2001 |isbn=3110167190}}
*{{cite book|last=Billingsley|first=Patrick|year=1995 |title=Probability and Measure|edition=3rd|publisher=John Wiley & Sons |isbn=0-471-00710-2}}
*{{cite journal|last=Bradley|first=Richard|title=Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions |journal=Probability Surveys|year=2005 |volume=2|pages=107–144 |arxiv=math/0511078 |doi=10.1214/154957805100000104 |bibcode=2005math.....11078B|s2cid=8395267}}
*{{cite book|last=Bradley|first=Richard|title=Introduction to Strong Mixing Conditions |edition=1st|year=2007|isbn=978-0-9740427-9-4 |publisher=Kendrick Press|location=Heber City, UT}}
*{{cite journal |last1=Dinov |first1=Ivo |last2=Christou |first2=Nicolas |last3=Sanchez |first3=Juana |year=2008 |title=Central Limit Theorem: New SOCR Applet and Demonstration Activity |journal=Journal of Statistics Education |publisher=ASA |volume=16 |issue=2 |pages=1–15 |url=http://www.amstat.org/publications/jse/v16n2/dinov.html |doi=10.1080/10691898.2008.11889560 |pmc=3152447 |pmid=21833159 |access-date=2008-08-23  |archive-date=2016-03-03  |archive-url=https://web.archive.org/web/20160303185802/http://www.amstat.org/publications/jse/v16n2/dinov.html |url-status=dead }}
*{{Cite book|last=Durrett|first=Richard|author-link=Rick Durrett|title=Probability: theory and examples|edition=3rd |year=2004 |publisher=Cambridge University Press|isbn=0521765390}}
*{{cite book|last=Fischer|first=Hans|year=2011|title=A History of the Central Limit Theorem: From Classical to Modern Probability Theory|series=Sources and Studies in the History of Mathematics and Physical Sciences |location=New York |publisher=Springer |isbn=978-0-387-87856-0 |doi=10.1007/978-0-387-87857-7 |zbl=1226.60004|mr=2743162 |url=http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf |archive-url=https://web.archive.org/web/20171031171033/http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf |archive-date=2017-10-31 |url-status=live}} 
*{{cite journal|last=Gaposhkin|first=V. F.|year=1966 |title=Lacunary series and independent functions |journal=Russian Mathematical Surveys|volume=21|issue=6 |pages=1–82|doi=10.1070/RM1966v021n06ABEH001196 |bibcode=1966RuMaS..21....1G|s2cid=250833638 }}.
*{{cite journal|last=Klartag |first=Bo'az |date=2007 |title=A central limit theorem for convex sets |journal=Inventiones Mathematicae |volume=168 |issue=1 |pages=91–131 |doi=10.1007/s00222-006-0028-8 |arxiv=math/0605014|bibcode=2007InMat.168...91K |s2cid=119169773}}
*{{cite journal|last=Klartag |first=Bo'az |date=2008 |title=A Berry–Esseen type inequality for convex bodies with an unconditional basis |journal=Probability Theory and Related Fields |doi=10.1007/s00440-008-0158-6 |arxiv=0705.0832 |volume=145 |issue=1–2 |pages=1–33 |s2cid=10163322}}

==External links==
{{Commons category}}
* [https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/central-limit-theorem Central Limit Theorem] at Khan Academy
*{{springer|title=Central limit theorem|id=p/c021180|mode=cs1}}
* {{MathWorld |title=Central Limit Theorem |urlname=CentralLimitTheorem}}
* [https://www.mctague.org/carl/blog/2021/04/23/central-limit-theorem/ A music video demonstrating the central limit theorem with a Galton board] by Carl McTague

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{{DEFAULTSORT:Central Limit Theorem}}
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[[Category:Theorems in probability theory]]
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