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| In [[probability theory]], the '''central limit theorem''' ('''CLT''') states that, given certain conditions, the [[arithmetic mean]] of a sufficiently large number of iterates of [[Statistical independence|independent]] [[random variables]], each with a well-defined expected value and well-defined [[variance]], will be approximately [[normal distribution|normally distributed]].<ref name=Rice/> That is, suppose that a [[Sample (statistics)|sample]] is obtained containing a large number of [[Random variate|observations]], each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be [[Probability distribution|distributed]] according to the [[normal distribution]] (commonly known as a "bell curve").
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| The central limit theorem has a number of variants. In its common form, the random variables must be [[identically distributed]]. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions.
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| In more general [[probability theory]], a '''central limit theorem''' is any of a set of [[Weak convergence of measures|weak-convergence]] theorems. They all express the fact that a sum of many [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of ''[[attractor]] distributions''. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with [[power law]] tail distributions decreasing as |''x''|<sup>−α−1</sup> where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-[[stable distribution]] with stability parameter (or index of stability) of α as the number of variables grows.<ref>{{Citation |first1=Johannes |last1=Voit |page=124 |year=2003|title=The Statistical Mechanics of Financial Markets|publisher=Springer-Verlag|isbn=3-540-00978-7}}</ref>
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| == Central limit theorems for independent sequences ==
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| [[File:Central limit thm.png|400px|thumb|right|A distribution being "smoothed out" by [[summation]], showing original [[Probability density function|density of distribution]] and three subsequent summations; see [[Illustration of the central limit theorem]] for further details.]]
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| === Classical CLT ===
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| Let {''X''<sub>1</sub>, ..., ''X<sub>n</sub>''} be a [[random sample]] of size ''n''—that is, a sequence of [[independent and identically distributed]] random variables drawn from distributions of [[expected value]]s given by µ and finite [[variance]]s given by σ<sup>2</sup>. Suppose we are interested in the [[sample mean|sample average]]
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| :<math>S_n := \frac{X_1+\cdots+X_n}{n}</math>
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| of these random variables. By the [[law of large numbers]], the sample averages [[Convergence of random variables#Convergence in probability|converge in probability]] and [[Convergence of random variables#Almost sure convergence|almost surely]] to the expected value µ as ''n'' → ∞. The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More precisely, it states that as ''n'' gets larger, the distribution of the difference between the sample average ''S<sub>n</sub>'' and its limit µ, when multiplied by the factor {{sqrt|''n''}} (that is {{sqrt|''n''}}(''S<sub>n</sub>'' − µ)), approximates the [[normal distribution]] with mean 0 and variance σ<sup>2</sup>. For large enough ''n,'' the distribution of ''S<sub>n</sub>'' is close to the normal distribution with mean µ and variance {{frac2|σ<sup>2</sup>|''n''}}. The usefulness of the theorem is that the distribution of {{sqrt|''n''}}(''S<sub>n</sub>'' − µ) approaches normality regardless of the shape of the distribution of the individual ''X<sub>i</sub>''’s. Formally, the theorem can be stated as follows:
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| <blockquote>'''Lindeberg–Lévy CLT.''' Suppose {''X''<sub>1</sub>, ''X''<sub>2</sub>, ...} is a sequence of [[independent and identically distributed|i.i.d.]] random variables with E[''X<sub>i</sub>''] = µ and Var[''X<sub>i</sub>''] = σ<sup>2</sup> < ∞. Then as ''n'' approaches infinity, the random variables {{sqrt|''n''}}(''S<sub>n</sub>'' − µ) [[convergence in distribution|converge in distribution]] to a [[normal distribution|normal]] ''N''(0, σ<sup>2</sup>):<ref>Billingsley (1995, p.357)</ref>
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| :<math>\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg) - \mu\bigg)\ \xrightarrow{d}\ N(0,\;\sigma^2).</math></blockquote>
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| In the case σ > 0, convergence in distribution means that the [[cumulative distribution function]]s of {{sqrt|''n''}}(''S<sub>n</sub>'' − µ) converge pointwise to the cdf of the N(0, σ<sup>2</sup>) distribution: for every real number ''z'',
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| :<math>\lim_{n\to\infty} \Pr[\sqrt{n}(S_n-\mu) \le z] = \Phi(z/\sigma),</math>
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| where Φ(''x'') is the standard normal cdf evaluated at ''x''. Note that the convergence is uniform in ''z'' in the sense that
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| :<math>\lim_{n\to\infty}\sup_{z\in{\mathbf R}}\bigl|\Pr[\sqrt{n}(S_n-\mu) \le z] - \Phi(z/\sigma)\bigr| = 0,</math>
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| where sup denotes the least upper bound (or [[supremum]]) of the set.<ref>Bauer (2001, Theorem 30.13, p.199)</ref>
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| === Lyapunov CLT ===
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| The theorem is named after Russian mathematician [[Aleksandr Lyapunov]]. In this variant of the central limit theorem the random variables ''X<sub>i</sub>'' have to be independent, but not necessarily identically distributed. The theorem also requires that random variables |''X<sub>i</sub>''| have [[moment (mathematics)|moment]]s of some order (2 + δ), and that the rate of growth of these moments is limited by the Lyapunov condition given below.
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| <blockquote>'''Lyapunov CLT.'''<ref>Billingsley (1995, p.362)</ref> Suppose {''X''<sub>1</sub>, ''X''<sub>2</sub>, ...} is a sequence of independent random variables, each with finite expected value μ<sub>''i''</sub> and variance {{SubSup|σ|''i''|2}}. Define
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| :<math>s_n^2 = \sum_{i=1}^n \sigma_i^2</math>
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| If for some δ > 0, the ''Lyapunov’s condition''
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| : <math>\lim_{n\to\infty} \frac{1}{s_{n}^{2+\delta}} \sum_{i=1}^{n} \operatorname{E}\big[\,|X_{i} - \mu_{i}|^{2+\delta}\,\big] = 0</math>
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| is satisfied, then a sum of (''X<sub>i</sub>'' − μ<sub>''i''</sub>)/''s<sub>n</sub>'' converges in distribution to a standard normal random variable, as ''n'' goes to infinity:
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| : <math>\frac{1}{s_n} \sum_{i=1}^{n} (X_i - \mu_i) \ \xrightarrow{d}\ \mathcal{N}(0,\;1).</math></blockquote>
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| In practice it is usually easiest to check the Lyapunov’s condition for δ = 1. If a sequence of random variables satisfies Lyapunov’s condition, then it also satisfies Lindeberg’s condition. The converse implication, however, does not hold.
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| === Lindeberg CLT ===
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| {{Main|Lindeberg's condition}}
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| 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).
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| Suppose that for every ε > 0
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| :<math> \lim_{n \to \infty} \frac{1}{s_n^2}\sum_{i = 1}^{n} \operatorname{E}\big[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\{ | X_i - \mu_i | > \varepsilon s_n \}} \big] = 0</math>
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| where '''1'''<sub>{…}</sub> is the [[indicator function]]. Then the distribution of the standardized sums <math>\frac{1}{s_n}\sum_{i = 1}^n \left( X_i - \mu_i \right)</math> converges towards the standard normal distribution N(0,1).
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| === Multidimensional CLT ===
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| Proofs that use characteristic functions can be extended to cases where each individual ''X''<sub>1</sub>, ..., ''X<sub>n</sub>'' is an independent and identically distributed [[random vector]] in '''R'''<sup>''k''</sup>, with mean vector μ = E(''X<sub>i</sub>'') and [[covariance matrix]] '''Σ''' (amongst the individual components of the vector). Now, if we take the summations of these vectors as being done componentwise, then the multidimensional central limit theorem states that when scaled, these converge to a [[multivariate normal distribution]].<ref>{{Citation|last = Van der Vaart|first = A. W.|title = Asymptotic statistics|year = 1998| publisher = Cambridge University Press | location = New York | isbn = 978-0-521-49603-2|lccn = QA276 .V22 1998| ref = CITEREFvan_der_Vaart1998}}</ref>
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| Let
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| :<math>\mathbf{X_i}=\begin{bmatrix} X_{i(1)} \\ \vdots \\ X_{i(k)} \end{bmatrix}</math>
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| be the ''i''-vector. The bold in '''X'''<sub>''i''</sub> means that it is a random vector, not a random (univariate) variable. Then the [[sum]] of the random vectors will be
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| :<math>\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} \left [ X_{i(1)} \right ] \\ \vdots \\ \sum_{i=1}^{n} \left [ X_{i(k)} \right ] \end{bmatrix} = \sum_{i=1}^{n} \left [ \mathbf{X_i} \right ]</math>
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| and the average will be
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| :<math>\left (\frac{1}{n}\right)\sum_{i=1}^{n} \left [ \mathbf{X_i} \right ]= \frac{1}{n}\begin{bmatrix} \sum_{i=1}^{n} \left [ X_{i(1)} \right ] \\ \vdots \\ \sum_{i=1}^{n} \left [ X_{i(k)} \right ] \end{bmatrix} = \begin{bmatrix} \bar X_{i(1)} \\ \vdots \\ \bar X_{i(k)} \end{bmatrix}=\mathbf{\bar X_n}</math>
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| and therefore
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| :<math>\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left [\mathbf{X_i} - E\left ( X_i\right ) \right ]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \left [ \mathbf{X_i} - \mu \right ]=\sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right) </math>.
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| The multivariate central limit theorem states that
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| :<math>\sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right)\ \stackrel{D}{\rightarrow}\ \mathcal{N}_k(0,\Sigma)</math>
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| where the [[covariance matrix]] Σ is equal to
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| :<math>\Sigma=\begin{bmatrix}
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| {Var \left (X_{1(1)} \right)} & {Cov \left (X_{1(1)},X_{1(2)} \right)} & Cov \left (X_{1(1)},X_{1(3)} \right) & \cdots & Cov \left (X_{1(1)},X_{1(k)} \right) \\
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| {Cov \left (X_{1(2)},X_{1(1)} \right)} & {Var \left (X_{1(2)} \right)} & {Cov \left(X_{1(2)},X_{1(3)} \right)} & \cdots & Cov \left(X_{1(2)},X_{1(k)} \right) \\
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| Cov \left (X_{1(3)},X_{1(1)} \right) & {Cov \left (X_{1(3)},X_{1(2)} \right)} & Var \left (X_{1(3)} \right) & \cdots & Cov \left (X_{1(3)},X_{1(k)} \right) \\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| Cov \left (X_{1(k)},X_{1(1)} \right) & Cov \left (X_{1(k)},X_{1(2)} \right) & Cov \left (X_{1(k)},X_{1(3)} \right) & \cdots & Var \left (X_{1(k)} \right) \\
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| \end{bmatrix}.</math>
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| == Central limit theorems for dependent processes ==
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| === CLT under weak dependence ===
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| 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 α(''n'') → 0 where α(''n'') is so-called [[Mixing (mathematics)#Mixing in stochastic processes|strong mixing coefficient]].
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| A simplified formulation of the central limit theorem under strong mixing is:<ref>Billingsley (1995, Theorem 27.4)</ref>
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| '''Theorem.''' Suppose that ''X''<sub>1</sub>, ''X''<sub>2</sub>, … is stationary and α-mixing with α<sub>''n''</sub> = ''O''(''n''<sup>−5</sup>) and that E(''X<sub>n</sub>'') = 0 and E(''X<sub>n</sub>''<sup>12</sup>) < ∞. Denote ''S<sub>n</sub>'' = ''X''<sub>1</sub> + … + ''X<sub>n</sub>'', then the limit
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| :<math> \sigma^2 = \lim_n \frac{E(S_n^2)}{n} </math>
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| exists, and if σ ≠ 0 then <math> S_n / (\sigma \sqrt n) </math> converges in distribution to ''N''(0, 1).
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| In fact,
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| :<math>\sigma^2 = E(X_1^2) + 2 \sum_{k=1}^{\infty} E(X_1 X_{1+k}),</math>
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| where the series converges absolutely.
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| The assumption σ ≠ 0 cannot be omitted, since the asymptotic normality fails for ''X<sub>n</sub>'' = ''Y<sub>n</sub>'' − ''Y''<sub>''n''−1</sub> where ''Y<sub>n</sub>'' are another [[stationary sequence]].
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| There is a stronger version of the theorem:<ref>Durrett (2004, Sect. 7.7(c), Theorem 7.8)</ref> the assumption E(''X<sub>n</sub>''<sup>12</sup>) < ∞ is replaced with {{nowrap|E({{!}}''X<sub>n</sub>''{{!}}<sup>2 + δ</sup>) < ∞, }} and the assumption α<sub>''n''</sub> = ''O''(''n''<sup>−5</sup>) is replaced with <math>\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.</math> Existence of such δ > 0 ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see {{harv|Bradley|2005}}.
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| === Martingale difference CLT ===
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| {{Main|Martingale central limit theorem}}
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| <blockquote>'''Theorem'''. Let a [[Martingale (probability theory)|martingale]] ''M<sub>n</sub>'' satisfy
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| * <math> \frac1n \sum_{k=1}^n \mathrm{E} ((M_k-M_{k-1})^2 | M_1,\dots,M_{k-1}) \to 1 </math> in probability as ''n'' tends to infinity,
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| * for every ε > 0, <math> \frac1n \sum_{k=1}^n \mathrm{E} \Big( (M_k-M_{k-1})^2; |M_k-M_{k-1}| > \varepsilon \sqrt n \Big) \to 0 </math> as ''n'' tends to infinity,
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| then <math> M_n / \sqrt n </math> converges in distribution to N(0,1) as ''n'' → ∞.<ref>Durrett (2004, Sect. 7.7, Theorem 7.4)</ref><ref>Billingsley (1995, Theorem 35.12)</ref></blockquote>
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| ''Caution:'' The [[restricted expectation]]{{clarify|reason=give source for this style of notation, else write using indicator functions|date=April 2012}} E(''X''; ''A'') should not be confused with the conditional expectation {{nowrap|E(''X''{{!}}''A'') {{=}} E(''X''; ''A'')/'''P'''(''A'').}}
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| == Remarks ==
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| === Proof of classical CLT ===
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| For a theorem of such fundamental importance to [[statistics]] and [[applied probability]], the central limit theorem has a remarkably simple proof using [[characteristic function (probability theory)|characteristic functions]]. It is similar to the proof of a (weak) [[Proof of the law of large numbers|law of large numbers]]. For any random variable, ''Y'', with zero [[mean]] and a unit variance (var(''Y'') = 1), the characteristic function of ''Y'' is, by [[Taylor's theorem]],
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| :<math>\varphi_Y(t) = 1 - {t^2 \over 2} + o(t^2), \quad t \rightarrow 0</math>
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| where ''o'' (''t''<sup>2</sup>) is "[[Big O notation|little o notation]]" for some function of ''t'' that goes to zero more rapidly than ''t''<sup>2</sup>.
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| Letting ''Y<sub>i</sub>'' be (''X<sub>i</sub>'' − μ)/σ, the standardized value of ''X<sub>i</sub>'', it is easy to see that the [[Standard_score#Standardizing_in_mathematical_statistics|standardized mean]] of the observations ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X<sub>n</sub>'' is
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| :<math>Z_n = \frac{n\overline{X}_n-n\mu}{\sigma \sqrt{n}} =\sum_{i=1}^n {Y_i \over \sqrt{n}}</math>
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| By simple properties of characteristic functions, the characteristic function of the sum is:
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| :<math>\varphi_{Z_n} =\varphi_{\sum_{i=1}^n {Y_i \over \sqrt{n}}}\left(t\right) = \varphi_{Y_1} \left(t / \sqrt{n} \right) \cdot \varphi_{Y_2} \left(t / \sqrt{n} \right)\cdot \ldots \cdot \varphi_{Y_n} \left(t / \sqrt{n} \right) = \left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n</math>
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| so that, by the limit of the [[exponential function]] ({{math| <var>e</var><sup>x</sup>{{=}} lim(1+x/n)<sup>n</sup>}}) the characteristic function of ''Z''<sub>''n''</sub> is
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| :<math>\left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n = \left[ 1 - {t^2 \over 2n} + o\left({t^2 \over n}\right) \right]^n \, \rightarrow \, e^{-t^2/2}, \quad n \rightarrow \infty.</math>
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| But this limit is just the characteristic function of a standard normal distribution N(0, 1), and the central limit theorem follows from the [[Lévy continuity theorem]], which confirms that the convergence of characteristic functions implies convergence in distribution.
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| ===Convergence to the limit===
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| 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.
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| If the third central [[Moment (mathematics)|moment]] E((''X''<sub>1</sub> − μ)<sup>3</sup>) exists and is finite, then the above convergence is [[uniform convergence|uniform]] and the speed of convergence is at least on the order of 1/''n''<sup>1/2</sup> (see [[Berry-Esseen theorem]]). [[Stein's method]]<ref name="stein1972">{{Citation| last = Stein |first=C. |authorlink=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| mr=402873 | zbl = 0278.60026| url=http://projecteuclid.org/euclid.bsmsp/1200514239
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| }}</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>{{Citation| title = Normal approximation by Stein's method| publisher = Springer| year = 2011| author = Chen, L.H.Y., Goldstein, L., and Shao, Q.M|isbn = 978-3-642-15006-7}}</ref>
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| The convergence to the normal distribution is monotonic, in the sense that the [[information entropy|entropy]] of ''Z''<sub>''n''</sub> increases [[monotonic function|monotonically]] to that of the normal distribution.<ref name=ABBN/>
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| 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 realisations of the sum of ''n'' independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as ''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.
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| ===Relation to the law of large numbers===
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| {{More footnotes|section|date=April 2012}}
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| [[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 behaviour of ''S''<sub>''n''</sub> as ''n'' approaches infinity?"{{Citation needed|date=April 2012}} In mathematical analysis, [[asymptotic series]] are one of the most popular tools employed to approach such questions.
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| Suppose we have an asymptotic expansion of ''f''(''n''):
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| : <math>f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \qquad (n \rightarrow \infty).</math>
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| Dividing both parts by φ<sub>1</sub>(''n'') and taking the limit will produce ''a''<sub>1</sub>, the coefficient of the highest-order term in the expansion, which represents the rate at which ''f''(''n'') changes in its leading term.
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| : <math>\lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.</math>
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| Informally, one can say: "''f''(''n'') grows approximately as ''a''<sub>1</sub> φ<sub>1</sub>(''n'')". Taking the difference between ''f''(''n'') and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about ''f''(''n''):
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| : <math>\lim_{n\to\infty}\frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)}=a_2 .</math>
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| Here one can say that the difference between the function and its approximation grows approximately as ''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.
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| Informally, something along these lines is happening when the sum, ''S<sub>n</sub>'', of independent identically distributed random variables, ''X''<sub>1</sub>, ..., ''X<sub>n</sub>'', is studied in classical probability theory.{{Citation needed|date=April 2012}} If each ''X<sub>i</sub>'' has finite mean μ, then by the law of large numbers, ''S<sub>n</sub>''/''n'' → μ.<ref>Rosenthal, Jeffrey Seth (2000) ''A first look at rigorous probability theory'', World Scientific, ISBN 981-02-4322-7.(Theorem 5.3.4, p. 47)</ref> If in addition each ''X<sub>i</sub>'' has finite variance σ<sup>2</sup>, then by the central limit theorem,
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| : <math> \frac{S_n-n\mu}{\sqrt{n}} \rightarrow \xi ,</math>
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| where ξ is distributed as N(0, σ<sup>2</sup>). This provides values of the first two constants in the informal expansion
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| : <math>S_n \approx \mu n+\xi \sqrt{n}. \, </math>
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| In the case where the ''X''<sub>''i''</sub>'s do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
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| :<math>\frac{S_n-a_n}{b_n} \rightarrow \Xi,</math>
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| or informally
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| : <math>S_n \approx a_n+\Xi b_n. \, </math>
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| Distributions Ξ which can arise in this way are called ''[[stable distribution|stable]]''.<ref>Johnson, Oliver Thomas (2004) ''Information theory and the central limit theorem'', Imperial College Press, 2004, ISBN 1-86094-473-6. (p. 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 ''b<sub>n</sub>'' may be proportional to ''n<sup>c</sup>'', for any ''c'' ≥ 1/2; it may also be multiplied by a [[slowly varying function]] of ''n''.<ref>Vladimir V. Uchaikin and V. M. Zolotarev (1999) ''Chance and stability: stable distributions and their applications'', VSP. ISBN 90-6764-301-7.(pp. 61–62)</ref><ref>Borodin, A. N. ; Ibragimov, Il'dar Abdulovich; Sudakov, V. N. (1995) ''Limit theorems for functionals of random walks'', AMS Bookstore, ISBN 0-8218-0438-3. (Theorem 1.1, p. 8 )</ref>
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| 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} </math> intermediate in size between n of the law of large numbers and √''n'' of the central limit theorem provides a non-trivial limiting behavior.
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| ===Alternative statements of the theorem===
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| ====Density functions====
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| 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,<ref>{{Citation|last=Petrov|first=V.V.|title=Sums of Independent Random Variables|year=1976|publisher=Springer-Verlag|location=New York-Heidelberg}}</ref> Chapter 7 for a particular local limit theorem for sums of i.i.d. random variables.
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| ====Characteristic functions====
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| 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. However, to state this more precisely, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
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| An equivalent statement can be made about [[Fourier transform]]s, since the characteristic function is essentially a Fourier transform.
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| ==Extensions to the theorem==
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| ===Products of positive random variables===
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| 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.
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| 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/>
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| ==Beyond the classical framework==
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| Asymptotic normality, that is, [[Convergence of random variables#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.
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| ===Convex body===
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| <blockquote>'''Theorem.''' There exists a sequence ε<sub>''n''</sub> ↓ 0 for which the following holds. Let ''n'' ≥ 1, and let random variables ''X''<sub>1</sub>, …, ''X<sub>n</sub>'' have a [[Logarithmically concave function|log-concave]] [[Probability_density_function#Probability functions associated with multiple variables|joint density]] ''f'' such that {{nowrap|''f''(''x''<sub>1</sub>, …, ''x<sub>n</sub>'') {{=}} ''f''({{!}}''x''<sub>1</sub>{{!}}, …, {{!}}''x<sub>n</sub>''{{!}})}} for all ''x''<sub>1</sub>, …, ''x<sub>n</sub>'', and E(''X<sub>k</sub>''<sup>2</sup>) = 1 for all ''k'' = 1, …, ''n''. Then the distribution of
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| :<math> \frac{X_1+\dots+X_n}{\sqrt n} </math>
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| is ε<sub>''n''</sub>-close to N(0, 1) in the [[Total variation distance of probability measures|total variation distance]].<ref>Klartag (2007, Theorem 1.2)</ref></blockquote>
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| These two 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.
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| 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".
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| Another example: {{nowrap|''f''(''x''<sub>1</sub>, …, ''x<sub>n</sub>'') {{=}} const · exp( − ({{!}}''x''<sub>1</sub>{{!}}<sup>α</sup> + … + {{!}}''x<sub>n</sub>''{{!}}<sup>α</sup>)<sup>β</sup>)}} where α > 1 and αβ > 1. If β = 1 then ''f''(''x''<sub>1</sub>, …, ''x<sub>n</sub>'') factorizes into {{nowrap|const · exp ( − {{!}}''x''<sub>1</sub>{{!}}<sup>α</sup>)…exp( − {{!}}''x<sub>n</sub>''{{!}}<sup>α</sup>), }} which means independence of ''X''<sub>1</sub>, …, ''X<sub>n</sub>''. In general, however, they are dependent.
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| The condition {{nowrap|''f''(''x''<sub>1</sub>, …, ''x<sub>n</sub>'') {{=}} ''f''({{!}}''x''<sub>1</sub>{{!}}, …, {{!}}''x<sub>n</sub>''{{!}})}} ensures that ''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.<ref>Durrett (2004, Section 2.4, Example 4.5)</ref>
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| Here is a [[Berry-Esseen theorem|Berry-Esseen]] type result.
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| '''Theorem.''' Let ''X''<sub>1</sub>, …, ''X<sub>n</sub>'' satisfy the assumptions of the previous theorem, then <ref>Klartag (2008, Theorem 1)</ref>
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| : <math> \bigg| \mathbb{P} \Big( a \le \frac{ X_1+\dots+X_n }{ \sqrt n } \le b \Big) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le \frac C n </math>
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| for all ''a'' < ''b''; here ''C'' is a [[mathematical constant|universal (absolute) constant]]. Moreover, for every ''c''<sub>1</sub>, …, ''c<sub>n</sub>'' ∈ '''R''' such that ''c''<sub>1</sub><sup>2</sup> + … + ''c<sub>n</sub>''<sup>2</sup> = 1,
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| : <math> \bigg| \mathbb{P} ( a \le c_1 X_1+\dots+c_n X_n \le b ) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le C ( c_1^4+\dots+c_n^4 ). </math>
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| The distribution of <math> (X_1+\dots+X_n)/\sqrt n </math> need not be approximately normal (in fact, it can be uniform).<ref>Klartag (2007, Theorem 1.1)</ref> However, the distribution of ''c''<sub>1</sub>''X''<sub>1</sub> + … + ''c<sub>n</sub>X<sub>n</sub>'' is close to ''N''(0, 1) (in the total variation distance) for most of vectors (''c''<sub>1</sub>, …, ''c<sub>n</sub>'') according to the uniform distribution on the sphere ''c''<sub>1</sub><sup>2</sup> + … + ''c<sub>n</sub>''<sup>2</sup> = 1.
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| ===Lacunary trigonometric series===
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| <blockquote>'''Theorem''' ([[Raphaël Salem|Salem]] - [[Antoni Zygmund|Zygmund]]). Let ''U'' be a random variable distributed uniformly on (0, 2π), and ''X<sub>k</sub>'' = ''r<sub>k</sub>'' cos(''n<sub>k</sub>U'' + ''a<sub>k</sub>''), where
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| * ''n''<sub>''k''</sub> satisfy the lacunarity condition: there exists ''q'' > 1 such that ''n''<sub>''k''+1</sub> ≥ ''qn''<sub>''k''</sub> for all ''k'',
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| * ''r''<sub>''k''</sub> are such that
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| :: <math> r_1^2 + r_2^2 + \cdots = \infty \text{ and } \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0, </math>
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| * 0 ≤ ''a''<sub>''k''</sub> < 2π.
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| Then<ref name=Zygmund/><ref>Gaposhkin (1966, Theorem 2.1.13)</ref>
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| : <math> \frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} } </math>
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| converges in distribution to ''N''(0, 1/2).</blockquote>
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| ===Gaussian polytopes===
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| <blockquote>'''Theorem''' Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> be independent random points on the plane '''R'''<sup>2</sup> each having the two-dimensional standard normal distribution. Let ''K''<sub>''n''</sub> be the [[convex hull]] of these points, and ''X<sub>n</sub>'' the area of ''K''<sub>''n''</sub> Then<ref>Barany & Vu (2007, Theorem 1.1)</ref>
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| : <math> \frac{ X_n - \mathrm{E} X_n }{ \sqrt{\operatorname{Var} X_n} } </math>
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| converges in distribution to ''N''(0, 1) as ''n'' tends to infinity.</blockquote>
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| The same holds in all dimensions (2, 3, ...).
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| The [[convex polytope|polytope]] ''K''<sub>''n''</sub> is called Gaussian random polytope.
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| A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.<ref>Barany & Vu (2007, Theorem 1.2)</ref>
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| ===Linear functions of orthogonal matrices===
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| A linear function of a matrix ''M'' is a linear combination of its elements (with given coefficients), ''M'' ↦ tr(''AM'') where ''A'' is the matrix of the coefficients; see [[Trace_(linear_algebra)#Inner product]].
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| A random [[orthogonal matrix]] is said to be distributed uniformly, if its distribution is the normalized [[Haar measure]] on the [[orthogonal group]] O(''n'', '''R'''); see [[Rotation matrix#Uniform random rotation matrices]].
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| '''Theorem.''' Let ''M'' be a random orthogonal ''n'' × ''n'' matrix distributed uniformly, and ''A'' a fixed ''n'' × ''n'' matrix such that tr(''AA*'') = ''n'', and let ''X'' = tr(''AM''). Then<ref name=Meckes/> the distribution of ''X'' is close to N(0, 1) in the total variation metric up to{{clarify|reason=what does up to mean here|date=June 2012}} 2{{sqrt|3}}/(''n''−1).
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| ===Subsequences===
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| '''Theorem.''' Let random variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, … ∈ ''L''<sub>2</sub>(Ω) be such that ''X<sub>n</sub>'' → 0 [[Weak convergence (Hilbert space)|weakly]] in ''L''<sub>2</sub>(Ω) and ''X<sub>n</sub>''<sup>2</sup> → 1 weakly in ''L''<sub>1</sub>(Ω). Then there exist integers ''n''<sub>1</sub> < ''n''<sub>2</sub> < … such that <math> ( X_{n_1}+\cdots+X_{n_k} ) / \sqrt k </math> converges in distribution to ''N''(0, 1) as ''k'' tends to infinity.<ref>Gaposhkin (1966, Sect. 1.5)</ref>
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| ===Tsallis statistics===
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| A generalization of the classical central limit theorem to the context of [[Tsallis statistics]] has been described by Umarov, Tsallis and Steinberg<ref name="Umarov2008">{{Citation |last1=Umarov |first1=Sabir |coauthors=Tsallis, Constantino and Steinberg, Stanly |year=2008 |title=On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics |journal=Milan j. Math. |volume=76 |issue= |pages=307–328 |publisher=Birkhauser Verlag |doi=10.1007/s00032-008-0087-y |url=http://www.cbpf.br/GrupPesq/StatisticalPhys/pdftheo/UmarovTsallisSteinberg2008.pdf |accessdate=2011-07-27 |postscript=.}}</ref> in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the ''q'' parameter, with independence being recovered as q->1. In analogy to the classical central limit theorem, such random variables with fixed mean and variance tend towards the [[q-Gaussian]] distribution, which maximizes the [[Tsallis entropy]] under these constraints. Umarov, Tsallis, Gell-Mann and Steinberg have defined similar generalizations of all symmetric alpha-[[stable distributions]], and have formulated a number of conjectures regarding their relevance to an even more general Central limit theorem.<ref name="Umarov2010">{{Citation |last1=Umarov |first1=Sabir |coauthors=Tsallis, Constantino, Gell-Mann, Murray and Steinberg, Stanly |year=2010 |title=Generalization of symmetric α-stable Lévy distributions for q>1 |journal=J Math Phys. |volume=51 |issue=3 |pages= 033502|publisher=American Institute of Physics |doi=10.1063/1.3305292 |pmc=2869267 |pmid=20596232 |postscript=.}}</ref>
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| ==Applications and examples==
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| ===Simple example===
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| [[File:Dice sum central limit theorem.svg|thumb|250px|Comparison of probability density functions, ''p''(''k'') for the sum of ''n'' fair 6-sided dice to show their convergence to a normal distribution with increasing ''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).]]
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| A simple example of the central limit theorem is rolling a large number of 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.
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| [[File:Empirical CLT - Figure - 040711.jpg|none|thumb|500px|This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 1 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/{{sqrt|''n''}}), which is called the standard deviation of the mean (since it refers to the spread of sample means).]]
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| ===Real applications===
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| [[File:UsaccHistogram.png|right|thumb|A histogram plot of monthly accidental deaths in the US, between 1973 and 1978 exhibits normality, due to the central limit theorem]]
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| Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem.<ref>Dinov, Christou & Sanchez (2008)</ref> One source<ref>[http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GCLT_Applications SOCR CLT Activity] wiki</ref> states the following examples:
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| *The probability distribution for total distance covered in a [[random walk]] (biased or unbiased) will tend toward a [[normal distribution]].
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| *Flipping a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).
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| From another viewpoint, the central limit theorem explains the common appearance of the "Bell Curve" in [[density estimation|density estimates]] applied to real world data. In cases like electronic noise, examination grades, and so on, we can often regard a single measured value as the weighted average of a large number of small effects. Using generalisations of the central limit theorem, we can then see that this would often (though not always) produce a final distribution that is approximately normal.
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| In general, the more a measurement is like the sum of independent variables with equal influence on the result, the more normality it exhibits. This justifies the common use of this distribution to stand in for the effects of unobserved variables in models like the [[linear model]].
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| ==Regression==
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| [[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 a large number of independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be assumed to be normally distributed.
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| ===Other illustrations===
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| {{Main|Illustration of the central limit theorem}}
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| 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">
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| Marasinghe, M., Meeker, W., Cook, D. & Shin, T.S.(1994 August),
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| "Using graphics and simulation to teach statistical concepts",
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| Paper presented at the Annual meeting of the
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| American Statistician Association, Toronto, Canada.
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| </ref>
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| ==History==
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| Tijms writes:<ref name=Tijms/>
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| {{quote|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.}}
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| Sir [[Francis Galton]] described the Central Limit Theorem as:<ref>Galton F. (1889) ''Natural Inheritance'' , [http://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm p. 66]</ref>
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| {{quote|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 marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.}}
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| 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>{{Citation|last=Pólya|first=George|authorlink=George Pólya|year=1920|title=Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem|journal=[[Mathematische Zeitschrift]]|volume=8|pages=171–181|language=German|url=http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0008|doi=10.1007/BF01206525|issue=3–4}}</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.
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| {{quote|text=The occurrence of the Gaussian probability density 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 Liapounoff. [...] }}
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| 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.<ref name=Fischer/> 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.
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| 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 never published.<ref>Hodges, Andrew (1983) ''Alan Turing: the enigma''. London: Burnett Books., pp. 87-88.{{full|date=November 2012}}</ref><ref name=Zabell/><ref name=Aldrich/>
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| ==See also==
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| * [[Asymptotic equipartition property]]
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| * [[Benford's law|Benford's Law]] - Result of extension of CLT to product of random variables.
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| * [[Central limit theorem for directional statistics]] - Central limit theorem applied to the case of directional statistics
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| * [[Delta method]] – to compute the limit distribution of a function of a random variable.
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| * [[Erdős-Kac theorem]] - connects the number of prime factors of an integer with the normal probability distribution
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| * [[Fisher–Tippett–Gnedenko theorem]] – limit theorem for extremum values (such as max{''X<sub>n</sub>''})
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| * [[Illustration of the central limit theorem]]
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| * [[Stable distribution]] - distributions such that linear combinations of i.i.d. samples lead to samples with the same distribution
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| * [[Theorem of de Moivre–Laplace]]
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| * [[Tweedie distribution|Tweedie convergence theorem]] - A theorem that can be considered to bridge between the central limit theorem and the Poisson convergence theorem<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = Chapman & Hall
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| | isbn = 978-0412997112
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| }}</ref>
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| ==Notes==
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| {{reflist|2|refs=
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| <ref name=ABBN>{{Citation| first1= S.|last1= Artstein | first2= K. |last2= Ball |first3= F. |last3= Barthe|first4= A. |last4= Naor|author4-link=Assaf Naor |year=2004 |url=http://www.ams.org/jams/2004-17-04/S0894-0347-04-00459-X/home.html |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 }}</ref>
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| <ref name=Aldrich>Aldrich, John (2009) "England and Continental Probability in the Inter-War Years", ''Electronic Journ@l for History of Probability and Statistics'', vol. 5/2, [http://www.jehps.net/decembre2009.html Decembre 2009]. (Section 3)</ref>
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| <ref name=Bernstein>[[Sergei Natanovich Bernstein|Bernstein, S.N.]] (1945) ''On the work of P.L.Chebyshev in Probability Theory'', Nauchnoe Nasledie P.L.Chebysheva. Vypusk Pervyi: Matematika. (Russian) [The Scientific Legacy of P. L. Chebyshev. First Part: Mathematics, Edited by S. N. Bernstein.] Academiya Nauk SSSR, Moscow-Leningrad, 174 pp.</ref>
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| <ref name=Fischer>{{citation|last=Fischer|first=Hans|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|year=2011|publisher=Springer|isbn=978-0-387-87856-0|doi=10.1007/978-0-387-87857-7|location=New York|zbl=1226.60004|mr=2743162}} (Chapter 2: The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods, Chapter 5.2: The Central Limit Theorem in the Twenties)</ref>
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| <ref name=LC1986>{{Citation| authorlink=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=10.2307/2245503}}</ref>
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| <ref name=Hald>Hald, Andreas [http://www.gbv.de/dms/goettingen/229762905.pdf ''A History of Mathematical Statistics from 1750 to 1930''], Ch.17. {{full|date=November 2012}}</ref>
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| <ref name=Meckes>{{Citation|last=Meckes|first= Elizabeth|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 }}</ref>
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| <ref name=Rempala>Rempala, G.; Wesolowski, J.(2002) [http://www.math.washington.edu/~ejpecp/EcpVol7/paper5.pdf "Asymptotics of products of sums and ''U''-statistics"], ''Electronic Communications in Probability'', 7, 47–54.</ref>
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| <ref name=Rice>{{Citation|last=Rice|first=John|title=Mathematical Statistics and Data Analysis|edition=Second|publisher=Duxbury Press|year=1995|isbn=0-534-20934-3}}){{page needed|date=April 2012}}</ref>
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| <ref name=Tijms>{{Citation | first=Tijms |last= Henk |year=2004 |title= Understanding Probability: Chance Rules in Everyday Life|location= Cambridge
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| |publisher= Cambridge University Press | isbn= 0-521-54036-4 | page=169}}</ref>
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| <ref name=Zabell>Zabell, S.L. (2005) ''Symmetry and its discontents: essays on the history of inductive probability'', Cambridge University Press. ISBN 0-521-44470-5. (pp. 199 ff.)</ref>
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| <ref name=Zygmund>{{citation|last=Zygmund|first=Antoni|authorlink=Antoni Zygmund|year=1959|title=Trigonometric series, Volume II|publisher=Cambridge}}. (2003 combined volume I,II: ISBN 0-521-89053-5) (Sect. XVI.5, Theorem 5-5)</ref>
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| }}
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| ==References==
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| *{{citation|last1=Barany|first1=Imre|authorlink1=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|doi=10.1214/009117906000000791 |arxiv=0610192}}
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| *{{citation|last=Bauer|first=Heinz|title=Measure and Integration Theory|publisher=de Gruyter|location=Berlin|year=2001|isbn=3110167190}}
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| *{{citation|last=Billingsley|first=Patrick|title=Probability and Measure|edition=Third|publisher=John Wiley & sons|year=1995|isbn=0-471-00710-2}}
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| *{{citation|last=Bradley|first=Richard|author-link=|title=Introduction to Strong Mixing Conditions |edition=First|year=2007|isbn=0-9740427-9-X|publisher=Kendrick Press|location=Heber City, UT}}
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| *{{citation|last=Bradley|first=Richard|author-link=|title=Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
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| |journal=Probability Surveys|year=2005|volume=2|pages=107–144 |arxiv=math/0511078v1 |doi=10.1214/154957805100000104 |url=http://arxiv.org/pdf/math/0511078.pdf}}
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| *{{citation|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 |url=http://www.amstat.org/publications/jse/v16n2/dinov.html }}
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| *{{Citation|last=Durrett|first=Richard|authorlink=Rick Durrett|title=Probability: theory and examples|edition=4th|year=2004|publisher= Cambridge University Press|
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| isbn=0521765390}}
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| *{{citation|last=Gaposhkin|first=V.F.|year=1966|title=Lacunary series and independent functions|journal=Russian Math. Surveys|volume=21|issue=6|pages=1–82|doi=10.1070/RM1966v021n06ABEH001196}}.
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| * <cite id=CITEREFKlartag2007>Klartag, Bo'az (2007), "A central limit theorem for convex sets", ''Inventiones Mathematicae'' '''168''', 91–131.{{doi|10.1007/s00222-006-0028-8}} Also [http://arxiv.org/abs/math/0605014 arXiv].</cite>
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| * <cite id=CITEREFKlartag2008>Klartag, Bo'az (2008), "A Berry-Esseen type inequality for convex bodies with an unconditional basis", ''Probability Theory and Related Fields''. {{doi|10.1007/s00440-008-0158-6}} Also [http://arxiv.org/abs/0705.0832 arXiv].</cite>
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| ==External links==
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| {{commons category}}
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| *{{springer|title=Central limit theorem|id=p/c021180}}
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| *[http://www.statisticalengineering.com/central_limit_theorem.html Animated examples of the CLT]
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| *[http://www.vias.org/simulations/simusoft_cenlimit.html Central Limit Theorem] interactive simulation to experiment with various parameters
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| *[http://ccl.northwestern.edu/curriculum/ProbLab/CentralLimitTheorem.html CLT in NetLogo (Connected Probability — ProbLab)] interactive simulation w/ a variety of modifiable parameters
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| *[http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem General Central Limit Theorem Activity] & corresponding [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR CLT Applet] (Select the Sampling Distribution CLT Experiment from the drop-down list of [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Experiments SOCR Experiments])
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| *[http://www.indiana.edu/~jkkteach/ExcelSampler/ Generate sampling distributions in Excel] Specify arbitrary population, [[sample size]], and sample statistic.
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| * MIT OpenCourseWare Lecture 18.440 ''Probability and Random Variables'', Spring 2011, Scott Sheffield [http://ocw.mit.edu/courses/mathematics/18-440-probability-and-random-variables-spring-2011/lecture-notes/MIT18_440S11_Lecture31.pdf Another proof.] Retrieved 2012-04-08.
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| *[http://www.causeweb.org CAUSEweb.org] is a site with many resources for teaching statistics including the Central Limit Theorem
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| * [http://demonstrations.wolfram.com/TheCentralLimitTheorem/ The Central Limit Theorem] by Chris Boucher, [[Wolfram Demonstrations Project]].
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| * {{MathWorld |title=Central Limit Theorem |urlname=CentralLimitTheorem}}
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| * [http://animation.yihui.name/prob:central_limit_theorem Animations for the Central Limit Theorem] by Yihui Xie using the [[R (programming language)|R]] package [http://cran.r-project.org/package=animation animation]
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| * Teaching demonstrations of the CLT: '''clt.examp''' function in {{cite manual|author=Greg Snow|year=2012|title= TeachingDemos: Demonstrations for teaching and learning. R package version 2.8.|url= http://CRAN.R-project.org/package=TeachingDemos|ref=harv}}
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| {{DEFAULTSORT:Central Limit Theorem}}
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| [[Category:Probability theorems]]
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| [[Category:Statistical theorems]]
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| [[Category:Statistical terminology]]
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| [[Category:Articles containing proofs]]
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| [[Category:Central limit theorem| ]]
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| [[Category:Asymptotic statistical theory]]
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