:Central limit theorem

File:IllustrationCentralTheorem.png

Let $\backslash \{X\_1,\; \backslash ldots,\; X\_n\}\backslash$ be a sequence of i.i.d. random variables having a distribution with expected value given by $\backslash mu$ and finite variance given by $\backslash sigma^2.$ Suppose we are interested in the sample average

$$\backslash bar\{X\}\_n\; \backslash equiv\; \backslash frac\{X\_1\; +\; \backslash cdots\; +\; X\_n\}\{n\}.$$

By the law of large numbers, the sample average converges almost surely (and therefore also converges in probability) to the expected value $\backslash mu$ as $n\backslash to\backslash infty.$

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

The usefulness of the theorem is that the distribution of $\backslash sqrt\{n\}(\backslash bar\{X\}\_n\; -\; \backslash mu)$ approaches normality regardless of the shape of the distribution of the individual $X\_i.$ Formally, the theorem can be stated as follows:

{{math theorem | name = Lindeberg–Lévy CLT | math_statement =

Suppose $X\_1,\; X\_2,\; X\_3\; \backslash ldots$ is a sequence of i.i.d. random variables with $\backslash operatorname\; E[X\_i]\; =\; \backslash mu$ and Then, as $n$ approaches infinity, the random variables $\backslash sqrt\{n\}(\backslash bar\{X\}\_n\; -\; \backslash mu)$ converge in distribution to a normal $\backslash mathcal\{N\}(0,\; \backslash sigma^2)$:{{sfnp|Billingsley|1995|p=357}}

$$\backslash sqrt\{n\}\backslash left(\backslash bar\{X\}\_n\; -\; \backslash mu\backslash right)\; \backslash mathrel\{\backslash overset\{d\}\{\backslash longrightarrow\}\}\; \backslash mathcal\{N\}\backslash left(0,\backslash sigma^2\backslash right)\; .$$}}

In the case $\backslash sigma\; >\; 0,$ convergence in distribution means that the cumulative distribution functions of $\backslash sqrt\{n\}(\backslash bar\{X\}\_n\; -\; \backslash mu)$ converge pointwise to the cdf of the $\backslash mathcal\{N\}(0,\; \backslash sigma^2)$ distribution: for every real number $z,$

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash mathbb\{P\}\backslash left[\backslash sqrt\{n\}(\backslash bar\{X\}\_n-\backslash mu)\; \backslash le\; z\backslash right]\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash mathbb\{P\}\backslash left[\backslash frac\{\backslash sqrt\{n\}(\backslash bar\{X\}\_n-\backslash mu)\}\{\backslash sigma\; \}\; \backslash le\; \backslash frac\{z\}\{\backslash sigma\}\backslash right]=$$

\Phi\left(\frac{z}{\sigma}\right) ,

where $\backslash Phi(z)$ is the standard normal cdf evaluated at $z.$ The convergence is uniform in $z$ in the sense that

$$\backslash lim\_\{n\backslash to\backslash infty\}\backslash ;\backslash sup\_\{z\backslash in\backslash R\}\backslash ;\backslash left|\backslash mathbb\{P\}\backslash left[\backslash sqrt\{n\}(\backslash bar\{X\}\_n-\backslash mu)\; \backslash le\; z\backslash right]\; -\; \backslash Phi\backslash left(\backslash frac\{z\}\{\backslash sigma\}\backslash right)\backslash right|\; =\; 0~,$$

where $\backslash sup$ denotes the least upper bound (or supremum) of the set.{{sfnp|Bauer|2001|loc=Theorem 30.13|p=199}}

In this variant of the central limit theorem the random variables $X\_i$ have to be independent, but not necessarily identically distributed. The theorem also requires that random variables $\backslash left|\; X\_i\backslash right|$ have moments of some order {{nowrap|$(2+\backslash delta)$,}} 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 $\backslash \{X\_1,\; \backslash ldots,\; X\_n,\; \backslash ldots\backslash \}$ is a sequence of independent random variables, each with finite expected value $\backslash mu\_i$ and variance {{nowrap|$\backslash sigma\_i^2$.}} Define

$$s\_n^2\; =\; \backslash sum\_\{i=1\}^n\; \backslash sigma\_i^2\; .$$

If for some {{nowrap|$\backslash delta\; >\; 0$,}} Lyapunov’s condition

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash ;\; \backslash frac\{1\}\{s\_\{n\}^\{2+\backslash delta\}\}\; \backslash ,\; \backslash sum\_\{i=1\}^\{n\}\; \backslash operatorname\; E\backslash left[\backslash left|X\_\{i\}\; -\; \backslash mu\_\{i\}\backslash right|^\{2+\backslash delta\}\backslash right]\; =\; 0$$

is satisfied, then a sum of $\backslash frac\{X\_i\; -\; \backslash mu\_i\}\{s\_n\}$ converges in distribution to a standard normal random variable, as $n$ goes to infinity:

$$\backslash frac\{1\}\{s\_n\}\backslash ,\backslash sum\_\{i=1\}^\{n\}\; \backslash left(X\_i\; -\; \backslash mu\_i\backslash right)\; \backslash mathrel\{\backslash overset\{d\}\{\backslash longrightarrow\}\}\; \backslash mathcal\{N\}(0,1)\; .$$}}

In practice it is usually easiest to check Lyapunov's condition for {{nowrap|$\backslash delta\; =\; 1$.}}

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

{{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 Lindeberg in 1920).

Suppose that for every $\backslash varepsilon\; >\; 0$,

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{s\_n^2\}\backslash sum\_\{i\; =\; 1\}^\{n\}\; \backslash operatorname\; E\backslash left[(X\_i\; -\; \backslash mu\_i)^2\; \backslash cdot\; \backslash mathbf\{1\}\_\{\backslash left\backslash \{\backslash left|\; X\_i\; -\; \backslash mu\_i\; \backslash right|\; >\; \backslash varepsilon\; s\_n\; \backslash right\backslash \}\}\; \backslash right]\; =\; 0$$

where $\backslash mathbf\{1\}\_\{\backslash \{\backslash ldots\backslash \}\}$ is the indicator function. Then the distribution of the standardized sums

$$\backslash frac\{1\}\{s\_n\}\backslash sum\_\{i\; =\; 1\}^n\; \backslash left(\; X\_i\; -\; \backslash mu\_i\; \backslash right)$$

converges towards the standard normal distribution {{nowrap|$\backslash mathcal\{N\}(0,\; 1)$.}}

Rather than summing an integer number $n$ of random variables and taking $n\; \backslash to\; \backslash infty$, the sum can be of a random number $N$ of random variables, with conditions on $N$.

{{math theorem | name = Robbins CLT{{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 }}{{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}} | math_statement =

Let $\backslash \{X\_i,\; i\; \backslash geq\; 1\backslash \}$ be independent, identically distributed random variables with $E(X\_i)\; =\; \backslash mu$ and $\backslash text\{Var\}(X\_i)\; =\; \backslash sigma^2$, and let $\backslash \{N\_n,\; n\; \backslash geq\; 1\backslash \}$ be a sequence of non-negative integer-valued random variables that are independent of $\backslash \{X\_i,\; i\; \backslash geq\; 1\backslash \}$. Assume for each $n\; =\; 1,\; 2,\; \backslash dots$ that and

\frac{N_n - E(N_n)}{\sqrt{\text{Var}(N_n)}} \xrightarrow{\quad d \quad} \mathcal{N}(0,1)

where $\backslash xrightarrow\{\backslash ,d\backslash ,\}$ denotes convergence in distribution and $\backslash mathcal\{N\}(0,1)$ is the normal distribution with mean 0, variance 1.

Then

\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)

}}

Proofs that use characteristic functions can be extended to cases where each individual $\backslash mathbf\{X\}\_i$ is a random vector in {{nowrap|$\backslash R^k$,}} with mean vector $\backslash boldsymbol\backslash mu\; =\; \backslash operatorname\; E[\backslash mathbf\{X\}\_i]$ and covariance matrix $\backslash mathbf\{\backslash Sigma\}$ (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.{{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}} Summation of these vectors is done component-wise.

For $i\; =\; 1,\; 2,\; 3,\; \backslash ldots,$ let

$$\backslash mathbf\{X\}\_i\; =\; \backslash begin\{bmatrix\}\; X\_\{i\}^\{(1)\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; X\_\{i\}^\{(k)\}\; \backslash end\{bmatrix\}$$

be independent random vectors. The sum of the random vectors $\backslash mathbf\{X\}\_1,\; \backslash ldots,\; \backslash mathbf\{X\}\_n$ is

$$\backslash sum\_\{i=1\}^\{n\}\; \backslash mathbf\{X\}\_i\; =\; \backslash begin\{bmatrix\}\; X\_\{1\}^\{(1)\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; X\_\{1\}^\{(k)\}$$

\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}

and their average is

$$\backslash mathbf\{\backslash bar\; X\_n\}\; =\; \backslash begin\{bmatrix\}\; \backslash bar\; X\_\{i\}^\{(1)\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash bar\; X\_\{i\}^\{(k)\}\; \backslash end\{bmatrix\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^\{n\}\; \backslash mathbf\{X\}\_i.$$

Therefore,

$$\backslash frac\{1\}\{\backslash sqrt\{n\}\}\; \backslash sum\_\{i=1\}^\{n\}\; \backslash left[\; \backslash mathbf\{X\}\_i\; -\; \backslash operatorname\; E\; \backslash left(\; \backslash mathbf\{X\}\_i\; \backslash right)\; \backslash right]\; =\; \backslash frac\{1\}\{\backslash sqrt\{n\}\}\backslash sum\_\{i=1\}^\{n\}\; (\; \backslash mathbf\{X\}\_i\; -\; \backslash boldsymbol\backslash mu\; )\; =\; \backslash sqrt\{n\}\backslash left(\backslash overline\{\backslash mathbf\{X\}\}\_n\; -\; \backslash boldsymbol\backslash mu\backslash right).$$

The multivariate central limit theorem states that

$$\backslash sqrt\{n\}\backslash left(\; \backslash overline\{\backslash mathbf\{X\}\}\_n\; -\; \backslash boldsymbol\backslash mu\; \backslash right)\; \backslash mathrel\{\backslash overset\{d\}\{\backslash longrightarrow\}\}\; \backslash mathcal\{N\}\_k(0,\backslash boldsymbol\backslash Sigma),$$

where the covariance matrix $\backslash boldsymbol\{\backslash Sigma\}$ is equal to

$$\backslash boldsymbol\backslash Sigma\; =\; \backslash begin\{bmatrix\}$$

{\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}~.

The multivariate central limit theorem can be proved using the Cramér–Wold theorem.

The rate of convergence is given by the following Berry–Esseen type result:

{{math theorem | name = Theorem{{cite web |first=Ryan |last=O’Donnell | author-link = Ryan O'Donnell (computer scientist) |year=2014 |title=Theorem 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 }} | math_statement =

Let $X\_1,\; \backslash dots,\; X\_n,\; \backslash dots$ be independent $\backslash R^d$-valued random vectors, each having mean zero. Write $S\; =\backslash sum^n\_\{i=1\}X\_i$ and assume $\backslash Sigma\; =\; \backslash operatorname\{Cov\}[S]$ is invertible. Let $Z\; \backslash sim\; \backslash mathcal\{N\}(0,\backslash Sigma)$ be a $d$-dimensional Gaussian with the same mean and same covariance matrix as $S$. Then for all convex sets {{nowrap|$U\; \backslash subseteq\; \backslash R^d$,}}

$$\backslash left|\backslash mathbb\{P\}[S\; \backslash in\; U]\; -\; \backslash mathbb\{P\}[Z\; \backslash in\; U]\backslash right|\; \backslash le\; C\; \backslash ,\; d^\{1/4\}\; \backslash gamma~,$$

where $C$ is a universal constant, {{nowrap|$\backslash gamma\; =\; \backslash sum^n\_\{i=1\}\; \backslash operatorname\; E\; \backslash left[\backslash left\backslash |\; \backslash Sigma^\{-1/2\}X\_i\backslash right\backslash |^3\_2\backslash right]$,}} and $\backslash |\backslash cdot\backslash |\_2$ denotes the Euclidean norm on {{nowrap|$\backslash R^d$.}}

}}

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

The generalized central limit theorem (GCLT) was an effort of multiple mathematicians (Bernstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937.{{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}} The first published complete proof of the GCLT was in 1937 by Paul Lévy in French.{{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}} An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.{{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}}

The statement of the GCLT is as follows:{{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}}

:A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X₁, X₂, X₃, ... and constants a_n > 0, b_n ∈ ℝ with

::a_n (X₁ + ... + X_n) − b_n → Z.

:Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy F_n(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.

A useful generalization of a sequence of independent, identically distributed random variables is a 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 strong mixing (also called α-mixing) defined by $\backslash alpha(n)\; \backslash to\; 0$ where $\backslash alpha(n)$ is so-called 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 $\backslash \{X\_1,\; \backslash ldots,\; X\_n,\; \backslash ldots\backslash \}$ is stationary and $\backslash alpha$-mixing with $\backslash alpha\_n\; =\; O\backslash left(n^\{-5\}\backslash right)$ and that $\backslash operatorname\; E[X\_n]\; =\; 0$ and {{nowrap|.}} Denote {{nowrap|$S\_n\; =\; X\_1\; +\; \backslash cdots\; +\; X\_n$,}} then the limit

$$\backslash sigma^2\; =\; \backslash lim\_\{n\backslash rightarrow\backslash infty\}\; \backslash frac\{\backslash operatorname\; E\backslash left(S\_n^2\backslash right)\}\{n\}$$

exists, and if $\backslash sigma\; \backslash ne\; 0$ then $\backslash frac\{S\_n\}\{\backslash sigma\backslash sqrt\{n\}\}$ converges in distribution to $\backslash mathcal\{N\}(0,\; 1)$.}}

In fact,

$$\backslash sigma^2\; =\; \backslash operatorname\; E\backslash left(X\_1^2\backslash right)\; +\; 2\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash operatorname\; E\backslash left(X\_1\; X\_\{1+k\}\backslash right),$$

where the series converges absolutely.

The assumption $\backslash sigma\; \backslash ne\; 0$ cannot be omitted, since the asymptotic normality fails for $X\_n\; =\; Y\_n\; -\; Y\_\{n-1\}$ where $Y\_n$ 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 is replaced with {{nowrap|,}} and the assumption $\backslash alpha\_n\; =\; O\backslash left(n^\{-5\}\backslash right)$ is replaced with

Existence of such $\backslash delta\; >\; 0$ ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see {{harv|Bradley|2007}}.

{{Main|Martingale central limit theorem}}

{{math theorem | math_statement = Let a martingale $M\_n$ satisfy

• $\backslash frac1n\; \backslash sum\_\{k=1\}^n\; \backslash operatorname\; E\backslash left[\backslash left(M\_k-M\_\{k-1\}\backslash right)^2\; \backslash mid\; M\_1,\backslash dots,M\_\{k-1\}\backslash right]\; \backslash to\; 1$ in probability as {{math|n → ∞}},
• for every {{math|ε > 0}}, $\backslash frac1n\; \backslash sum\_\{k=1\}^n\{\backslash operatorname\; E\backslash left[\backslash left(M\_k-M\_\{k-1\}\; \backslash right)^2\backslash mathbf\{1\}\backslash left[|M\_k-M\_\{k-1\}|>\backslash varepsilon\backslash sqrt\{n\}\backslash right]\backslash right]\}\; \backslash to\; 0$ as {{math|n → ∞}},

then $\backslash frac\{M\_n\}\{\backslash sqrt\{n\}\}$ converges in distribution to $\backslash mathcal\{N\}(0,\; 1)$ as $n\; \backslash to\; \backslash infty$.{{sfnp|Durrett|2004|loc=Sect. 7.7, Theorem 7.4}}{{sfnp|Billingsley|1995 |loc=Theorem 35.12}}}}

The central limit theorem has a proof using characteristic functions.{{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 }} It is similar to the proof of the (weak) law of large numbers.

Assume $\backslash \{X\_1,\; \backslash ldots,\; X\_n,\; \backslash ldots\; \backslash \}$ are independent and identically distributed random variables, each with mean $\backslash mu$ and finite variance {{nowrap|$\backslash sigma^2$.}} The sum $X\_1\; +\; \backslash cdots\; +\; X\_n$ has mean $n\backslash mu$ and variance {{nowrap|$n\backslash sigma^2$.}} Consider the random variable

$$Z\_n\; =\; \backslash frac\{X\_1+\backslash cdots+X\_n\; -\; n\; \backslash mu\}\{\backslash sqrt\{n\; \backslash sigma^2\}\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{X\_i\; -\; \backslash mu\}\{\backslash sqrt\{n\; \backslash sigma^2\}\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{\backslash sqrt\{n\}\}\; Y\_i,$$

where in the last step we defined the new random variables {{nowrap|$Y\_i\; =\; \backslash frac\{X\_i\; -\; \backslash mu\}\{\backslash sigma\}$,}} each with zero mean and unit variance {{nowrap|($\backslash operatorname\{var\}(Y)\; =\; 1$).}} The characteristic function of $Z\_n$ is given by

$$\backslash varphi\_\{Z\_n\}\backslash !(t)\; =\; \backslash varphi\_\{\backslash sum\_\{i=1\}^n\; \{\backslash frac\{1\}\{\backslash sqrt\{n\}\}Y\_i\}\}\backslash !(t)\; \backslash \; =\backslash \; \backslash varphi\_\{Y\_1\}\backslash !\backslash !\backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\; \backslash varphi\_\{Y\_2\}\backslash !\backslash !\; \backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\backslash cdots\; \backslash varphi\_\{Y\_n\}\backslash !\backslash !\; \backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\; \backslash \; =\backslash \; \backslash left[\backslash varphi\_\{Y\_1\}\backslash !\backslash !\backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\backslash right]^n,$$

where in the last step we used the fact that all of the $Y\_i$ are identically distributed. The characteristic function of $Y\_1$ is, by Taylor's theorem,

$$\backslash varphi\_\{Y\_1\}\backslash !\backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\; =\; 1\; -\; \backslash frac\{t^2\}\{2n\}\; +\; o\backslash !\backslash left(\backslash frac\{t^2\}\{n\}\backslash right),\; \backslash quad\; \backslash left(\backslash frac\{t\}\{\backslash sqrt\{n\}\}\backslash right)\; \backslash to\; 0$$

where $o(t^2\; /\; n)$ is "Little-o notation" for some function of $t$ that goes to zero more rapidly than {{nowrap|$t^2\; /\; n$.}} By the limit of the exponential function {{nowrap|($e^x\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left(1\; +\; \backslash frac\{x\}\{n\}\backslash right)^n$),}} the characteristic function of $Z\_n$ equals

$$\backslash varphi\_\{Z\_n\}(t)\; =\; \backslash left(1\; -\; \backslash frac\{t^2\}\{2n\}\; +\; o\backslash left(\backslash frac\{t^2\}\{n\}\backslash right)\; \backslash right)^n\; \backslash rightarrow\; e^\{-\backslash frac\{1\}\{2\}\; t^2\},\; \backslash quad\; n\; \backslash to\; \backslash infty.$$

All of the higher order terms vanish in the limit {{nowrap|$n\backslash to\backslash infty$.}} The right hand side equals the characteristic function of a standard normal distribution $\backslash mathcal\{N\}(0,\; 1)$, which implies through Lévy's continuity theorem that the distribution of $Z\_n$ will approach $\backslash mathcal\{N\}(0,1)$ as {{nowrap|$n\backslash to\backslash infty$.}} Therefore, the sample average

$$\backslash bar\{X\}\_n\; =\; \backslash frac\{X\_1+\backslash cdots+X\_n\}\{n\}$$

is such that

$$\backslash frac\{\backslash sqrt\{n\}\}\{\backslash sigma\}(\backslash bar\{X\}\_n\; -\; \backslash mu)\; =\; Z\_n$$

converges to the normal distribution {{nowrap|$\backslash mathcal\{N\}(0,\; 1)$,}} from which the central limit theorem follows.

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 because the limiting cumulative distribution function is continuous. If the third central moment $\backslash operatorname\{E\}\backslash left[(X\_1\; -\; \backslash mu)^3\backslash right]$ exists and is finite, then the speed of convergence is at least on the order of $1\; /\; \backslash sqrt\{n\}$ (see Berry–Esseen theorem). Stein's method{{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 }} can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.{{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}}

The convergence to the normal distribution is monotonic, in the sense that the entropy of $Z\_n$ increases monotonically to that of the normal distribution.

The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables 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.

Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.{{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 }}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{{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 }} 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,{{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 }} 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.

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_{{mvar|n}}}} 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 $f(n)$:

$$f(n)=\; a\_1\; \backslash varphi\_\{1\}(n)+a\_2\; \backslash varphi\_\{2\}(n)+O\backslash big(\backslash varphi\_\{3\}(n)\backslash big)\; \backslash qquad\; (n\; \backslash to\; \backslash infty).$$

Dividing both parts by {{math|φ₁(n)}} and taking the limit will produce {{math|a₁}}, the coefficient of the highest-order term in the expansion, which represents the rate at which {{math|f(n)}} changes in its leading term.

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{f(n)\}\{\backslash varphi\_\{1\}(n)\}\; =\; a\_1.$$

Informally, one can say: "{{math|f(n)}} grows approximately as {{math|a₁φ₁(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)}}:

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{f(n)-a\_1\; \backslash varphi\_\{1\}(n)\}\{\backslash varphi\_\{2\}(n)\}\; =\; a\_2\; .$$

Here one can say that the difference between the function and its approximation grows approximately as {{math|a₂φ₂(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_n}}, of independent identically distributed random variables, {{math|X₁, ..., X_n}}, is studied in classical probability theory.{{Citation needed|date=April 2012}} If each {{mvar|X_i}} has finite mean {{mvar|μ}}, then by the law of large numbers, {{math|{{sfrac|S_n|n}} → μ}}.{{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}} If in addition each {{mvar|X_i}} has finite variance {{math|σ²}}, then by the central limit theorem,

$$\backslash frac\{S\_n-n\backslash mu\}\{\backslash sqrt\{n\}\}\; \backslash to\; \backslash xi\; ,$$

where {{mvar|ξ}} is distributed as {{math|N(0,σ²)}}. This provides values of the first two constants in the informal expansion

$$S\_n\; \backslash approx\; \backslash mu\; n+\backslash xi\; \backslash sqrt\{n\}.$$

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

$$\backslash frac\{S\_n-a\_n\}\{b\_n\}\; \backslash rightarrow\; \backslash Xi,$$

or informally

$$S\_n\; \backslash approx\; a\_n+\backslash Xi\; b\_n.$$

Distributions {{math|Ξ}} which can arise in this way are called stable.{{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}} 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_n}} may be proportional to {{mvar|n^c}}, for any {{math|c ≥ {{sfrac|1|2}}}}; it may also be multiplied by a slowly varying function of {{mvar|n}}.{{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}}{{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}}

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.

The 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{{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}} for a particular local limit theorem for sums of independent and identically distributed random variables.

Since the 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 transforms, since the characteristic function is essentially a Fourier transform.

Let {{mvar|S_n}} be the sum of {{mvar|n}} random variables. Many central limit theorems provide conditions such that {{math|{{mvar|S_n}}/{{sqrt|Var({{mvar|S_n}})}}}} 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|σ²}} and function {{mvar|f(n)}} such that {{math|{{mvar|S_n}}/(σ{{sqrt|{{mvar|n⋅f}}({{mvar|n}})}})}} converges in distribution to {{math|N(0,1)}} as {{math|{{mvar|n}}→ ∞}}.

{{math theorem | name = Lemma{{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}} | math_statement = Suppose $X\_1,\; X\_2,\; \backslash dots$ is a sequence of real-valued and strictly stationary random variables with $\backslash operatorname\; E(X\_i)\; =\; 0$ for all {{nowrap|$i$,}} {{nowrap|$g\; :\; [0,1]\; \backslash to\; \backslash R$,}} and {{nowrap|$S\_n\; =\; \backslash sum\_\{i=1\}^\{n\}\; g\backslash left(\backslash tfrac\{i\}\{n\}\backslash right)\; X\_i$.}} Construct

$$\backslash sigma^2\; =\; \backslash operatorname\; E(X\_1^2)\; +\; 2\backslash sum\_\{i=1\}^\backslash infty\; \backslash operatorname\; E(X\_1\; X\_\{1+i\})$$

1. If $\backslash sum\_\{i=1\}^\backslash infty\; \backslash operatorname\; E(X\_1\; X\_\{1+i\})$ is absolutely convergent, , and then $\backslash mathrm\{Var\}(S\_n)/(n\; \backslash gamma\_n)\; \backslash to\; \backslash sigma^2$ as $n\; \backslash to\; \backslash infty$ where {{nowrap|$\backslash gamma\_n\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{n\}\; \backslash left(g\backslash left(\backslash tfrac\{i\}\{n\}\backslash right)\backslash right)^2$.}}
2. If in addition $\backslash sigma\; >\; 0$ and $S\_n/\backslash sqrt\{\backslash mathrm\{Var\}(S\_n)\}$ converges in distribution to $\backslash mathcal\{N\}(0,1)$ as $n\; \backslash to\; \backslash infty$ then $S\_n/(\backslash sigma\backslash sqrt\{n\; \backslash gamma\_n\})$ also converges in distribution to $\backslash mathcal\{N\}(0,1)$ as {{nowrap|$n\; \backslash to\; \backslash infty$.}}

}}

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.

Asymptotic normality, that is, 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.

{{math theorem | math_statement = There exists a sequence {{math|ε_n ↓ 0}} for which the following holds. Let {{math|n ≥ 1}}, and let random variables {{math|X₁, ..., X_n}} have a log-concave joint density {{mvar|f}} such that {{math|1=f(x₁, ..., x_n) = f({{abs|x₁}}, ..., {{abs|x_n}})}} for all {{math|x₁, ..., x_n}}, and {{math|1=E(X{{su|b=k|p=2}}) = 1}} for all {{math|1=k = 1, ..., n}}. Then the distribution of

$$\backslash frac\{X\_1+\backslash cdots+X\_n\}\{\backslash sqrt\; n\}$$

is {{mvar|ε_n}}-close to $\backslash mathcal\{N\}(0,\; 1)$ in the total variation distance.{{sfnp|Klartag|2007|loc=Theorem 1.2}}}}

These two {{mvar|ε_n}}-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(x₁, ..., x_n) = const · exp(−({{abs|x₁}}^α + ⋯ + {{abs|x_n}}^α)^β)}} where {{math|α > 1}} and {{math|αβ > 1}}. If {{math|1=β = 1}} then {{math|f(x₁, ..., x_n)}} factorizes into {{math|const · exp (−{{abs|x₁}}^α) … exp(−{{abs|x_n}}^α), }} which means {{math|X₁, ..., X_n}} are independent. In general, however, they are dependent.

The condition {{math|1=f(x₁, ..., x_n) = f({{abs|x₁}}, ..., {{abs|x_n}})}} ensures that {{math|X₁, ..., X_n}} are of zero mean and uncorrelated;{{Citation needed|date=June 2012}} still, they need not be independent, nor even 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 type result.

{{math theorem | math_statement = Let {{math|X₁, ..., X_n}} satisfy the assumptions of the previous theorem, then{{Sfnp|Klartag|2008|loc=Theorem 1}}

$$\backslash left|\; \backslash mathbb\{P\}\; \backslash left(\; a\; \backslash le\; \backslash frac\{\; X\_1+\backslash cdots+X\_n\; \}\{\; \backslash sqrt\; n\; \}\; \backslash le\; b\; \backslash right)\; -\; \backslash frac1\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_a^b\; e^\{-\backslash frac\{1\}\{2\}\; t^2\}\; \backslash ,\; dt\; \backslash right|\; \backslash le\; \backslash frac\{C\}\{n\}$$

for all {{math|a < b}}; here {{mvar|C}} is a universal (absolute) constant. Moreover, for every {{math|c₁, ..., c_n ∈ R}} such that {{math|1=c{{su|b=1|p=2}} + ⋯ + c{{su|b=n|p=2}} = 1}},

$$\backslash left|\; \backslash mathbb\{P\}\; \backslash left(\; a\; \backslash le\; c\_1\; X\_1+\backslash cdots+c\_n\; X\_n\; \backslash le\; b\; \backslash right)\; -\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_a^b\; e^\{-\backslash frac\{1\}\{2\}\; t^2\}\; \backslash ,\; dt\; \backslash right|\; \backslash le\; C\; \backslash left(\; c\_1^4+\backslash dots+c\_n^4\; \backslash right).$$}}

The distribution of {{math|{{sfrac|X₁ + ⋯ + X_n|{{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|c₁X₁ + ⋯ + c_nX_n}} is close to $\backslash mathcal\{N\}(0,\; 1)$ (in the total variation distance) for most vectors {{math|(c₁, ..., c_n)}} according to the uniform distribution on the sphere {{math|1=c{{su|b=1|p=2}} + ⋯ + c{{su|b=n|p=2}} = 1}}.

{{math theorem | name = Theorem (Salem–Zygmund) | math_statement =

Let {{mvar|U}} be a random variable distributed uniformly on {{math|(0,2π)}}, and {{math|1=X_k = r_k cos(n_kU + a_k)}}, where

• {{mvar|n_k}} satisfy the lacunarity condition: there exists {{math|q > 1}} such that {{math|n_{k + 1} ≥ qn_k}} for all {{mvar|k}},
• {{mvar|r_k}} are such that
  $$r\_1^2\; +\; r\_2^2\; +\; \backslash cdots\; =\; \backslash infty\; \backslash quad\backslash text\{\; and\; \}\backslash quad\; \backslash frac\{\; r\_k^2\; \}\{\; r\_1^2+\backslash cdots+r\_k^2\; \}\; \backslash to\; 0,$$
• {{math|0 ≤ a_k < 2π}}.

Then{{sfnp|Gaposhkin|1966|loc=Theorem 2.1.13}}

$$\backslash frac\{\; X\_1+\backslash cdots+X\_k\; \}\{\; \backslash sqrt\{r\_1^2+\backslash cdots+r\_k^2\}\; \}$$

converges in distribution to $\backslash mathcal\{N\}\backslash big(0,\; \backslash frac\{1\}\{2\}\backslash big)$.}}

{{math theorem | math_statement =

Let {{math|A₁, ..., A_n}} be independent random points on the plane {{math|R²}} each having the two-dimensional standard normal distribution. Let {{mvar|K_n}} be the convex hull of these points, and {{mvar|X_n}} the area of {{mvar|K_n}} Then{{sfnp|Bárány|Vu|2007|loc=Theorem 1.1}}

$$\backslash frac\{\; X\_n\; -\; \backslash operatorname\; E\; (X\_n)\; \}\{\; \backslash sqrt\{\backslash operatorname\{Var\}\; (X\_n)\}\; \}$$

converges in distribution to $\backslash mathcal\{N\}(0,\; 1)$ as {{mvar|n}} tends to infinity.}}

The same also holds in all dimensions greater than 2.

The polytope {{mvar|K_n}} 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}}

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 the distribution of {{mvar|X}} is close to $\backslash mathcal\{N\}(0,\; 1)$ 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}}}}.}}

{{math theorem | math_statement = Let random variables {{math|X₁, X₂, ... ∈ L₂(Ω)}} be such that {{math|X_n → 0}} weakly in {{math|L₂(Ω)}} and {{math|X{{su|b=n|2}} → 1}} weakly in {{math|L₁(Ω)}}. Then there exist integers {{math|n₁ < n₂ < ⋯}} such that

$$\backslash frac\{\; X\_\{n\_1\}+\backslash cdots+X\_\{n\_k\}\; \}\{\; \backslash sqrt\; k\; \}$$

converges in distribution to $\backslash mathcal\{N\}(0,\; 1)$ as {{mvar|k}} tends to infinity.{{sfnp|Gaposhkin|1966|loc=Sect. 1.5}}}}

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.{{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}}{{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}}

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 statistics 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 chi-squared values that quantify the goodness of the fit (the fit is good if the reduced 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

Regression analysis, and in particular ordinary least squares, specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive 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.

{{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.{{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}}

Dutch mathematician Henk Tijms writes:

{{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:{{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}}

{{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.{{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}} 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". The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya in 1920 translates as follows.

{{blockquote|text=The occurrence of the Gaussian probability density {{math|1 {{=}} e^−x2}} 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. ... }}

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald. Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by von Mises, Pólya, Lindeberg, Lévy, and 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. 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 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.{{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 }}

• 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_n}}})
• Irwin–Hall distribution
• Markov chain central limit theorem
• Normal distribution
• Tweedie convergence theorem – a theorem that can be considered to bridge between the central limit theorem and the Poisson convergence theorem{{cite book| last= Jørgensen|first= Bent | year = 1997| title = The Theory of Dispersion Models| publisher = Chapman & Hall | isbn = 978-0412997112}}
• Donsker's theorem

{{Reflist|30em|refs=

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{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

}}

• {{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}}
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• {{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 }}
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• {{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}}

{{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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