In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a welldefined expected value and welldefined variance, will be approximately normally distributed, regardless of the underlying distribution.^{[1]}^{[2]} That is, suppose that a sample is obtained containing a large number of 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 distributed according to the normal distribution (commonly known as a "bell curve").
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 nonidentical distributions or for nonindependent observations, given that they comply with certain conditions.
In more general probability theory, a central limit theorem is any of a set of weakconvergence theorems. They all express the fact that a sum of many 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^{−α−1} where 0 < α < 2 (and therefore having infinite variance) will tend to an alphastable distribution with stability parameter (or index of stability) of α as the number of variables grows.^{[3]}
Contents

Central limit theorems for independent sequences 1

Classical CLT 1.1

Lyapunov CLT 1.2

Lindeberg CLT 1.3

Multidimensional CLT 1.4

Central limit theorems for dependent processes 2

CLT under weak dependence 2.1

Martingale difference CLT 2.2

Remarks 3

Proof of classical CLT 3.1

Convergence to the limit 3.2

Relation to the law of large numbers 3.3

Alternative statements of the theorem 3.4

Density functions 3.4.1

Characteristic functions 3.4.2

Extensions to the theorem 4

Products of positive random variables 4.1

Beyond the classical framework 5

Convex body 5.1

Lacunary trigonometric series 5.2

Gaussian polytopes 5.3

Linear functions of orthogonal matrices 5.4

Subsequences 5.5

Tsallis statistics 5.6

Random walk on a crystal lattice 5.7

Applications and examples 6

Simple example 6.1

Real applications 6.2

Regression 7

History 8

See also 9

Notes 10

References 11

External links 12
Central limit theorems for independent sequences
Classical CLT
Let {X_{1}, ..., X_{n}} be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ^{2}. Suppose we are interested in the sample average

S_n := \frac{X_1+\cdots+X_n}{n}
of these random variables. By the law of large numbers, the sample averages converge in probability and 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_{n} and its limit µ, when multiplied by the factor √n (that is √n(S_{n} − µ)), approximates the normal distribution with mean 0 and variance σ^{2}. For large enough n, the distribution of S_{n} is close to the normal distribution with mean µ and variance σ^{2}/n. The usefulness of the theorem is that the distribution of √n(S_{n} − µ) approaches normality regardless of the shape of the distribution of the individual X_{i}’s. Formally, the theorem can be stated as follows:
Lindeberg–Lévy CLT. Suppose {X_{1}, X_{2}, ...} is a sequence of i.i.d. random variables with E[X_{i}] = µ and Var[X_{i}] = σ^{2} < ∞. Then as n approaches infinity, the random variables √n(S_{n} − µ) converge in distribution to a normal N(0, σ^{2}):^{[4]}

\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg)  \mu\bigg)\ \xrightarrow{d}\ N(0,\;\sigma^2).
In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n(S_{n} − µ) converge pointwise to the cdf of the N(0, σ^{2}) distribution: for every real number z,

\lim_{n\to\infty} \Pr[\sqrt{n}(S_n\mu) \le z] = \Phi(z/\sigma),
where Φ(x) is the standard normal cdf evaluated at x. Note that the convergence is uniform in z in the sense that

\lim_{n\to\infty}\sup_{z\in{\mathbf R}}\bigl\Pr[\sqrt{n}(S_n\mu) \le z]  \Phi(z/\sigma)\bigr = 0,
where sup denotes the least upper bound (or supremum) of the set.^{[5]}
Lyapunov CLT
The theorem is named after Russian mathematician Aleksandr Lyapunov. 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 X_{i} have moments of some order (2 + δ), and that the rate of growth of these moments is limited by the Lyapunov condition given below.
Lyapunov CLT.^{[6]} Suppose {X_{1}, X_{2}, ...} is a sequence of independent random variables, each with finite expected value μ_{i} and variance σ 2
i . Define

s_n^2 = \sum_{i=1}^n \sigma_i^2
If for some δ > 0, the Lyapunov’s condition

\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
is satisfied, then a sum of (X_{i} − μ_{i})/s_{n} converges in distribution to a standard normal random variable, as n goes to infinity:

\frac{1}{s_n} \sum_{i=1}^{n} (X_i  \mu_i) \ \xrightarrow{d}\ \mathcal{N}(0,\;1).
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.
Lindeberg CLT
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 ε > 0

\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
where 1_{{...}} is the indicator function. Then the distribution of the standardized sums \frac{1}{s_n}\sum_{i = 1}^n \left( X_i  \mu_i \right) converges towards the standard normal distribution N(0,1).
Multidimensional CLT
Proofs that use characteristic functions can be extended to cases where each individual X_{i} is a random vector in R^{k}, with mean vector μ = E(X_{i}) and covariance matrix Σ (amongst the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done componentwise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.^{[7]}
Let

\mathbf{X_i}=\begin{bmatrix} X_{i(1)} \\ \vdots \\ X_{i(k)} \end{bmatrix}
be the kvector. The bold in X_{i} means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be

\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} \mathbf{X_i}
and the average is

\frac{1}{n} \sum_{i=1}^{n} \mathbf{X_i}= \frac{1}{n}\begin{bmatrix} \sum_{i=1}^{n} X_{i(1)} \\ \vdots \\ \sum_{i=1}^{n} X_{i(k)} \end{bmatrix} = \begin{bmatrix} \bar X_{i(1)} \\ \vdots \\ \bar X_{i(k)} \end{bmatrix}=\mathbf{\bar X_n}
and therefore

\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} ( \mathbf{X_i}  \mu ) = \sqrt{n}\left(\mathbf{\overline{X}}_n  \mu\right) .
The multivariate central limit theorem states that

\sqrt{n}\left(\mathbf{\overline{X}}_n  \mu\right)\ \stackrel{D}{\rightarrow}\ \mathcal{N}_k(0,\Sigma)
where the covariance matrix Σ is equal to

\Sigma=\begin{bmatrix} {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) \\ {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) \\ 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) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 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) \\ \end{bmatrix}.
Central limit theorems for dependent processes
CLT under weak dependence
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 αmixing) defined by α(n) → 0 where α(n) is socalled strong mixing coefficient.
A simplified formulation of the central limit theorem under strong mixing is:^{[8]}
Theorem. Suppose that X_{1}, X_{2}, ... is stationary and αmixing with α_{n} = O(n^{−5}) and that E(X_{n}) = 0 and E(X_{n}^{2}) < ∞. Denote S_{n} = X_{1} + ... + X_{n}, then the limit

\sigma^2 = \lim_n \frac{E(S_n^2)}{n}
exists, and if σ ≠ 0 then S_n / (\sigma \sqrt n) converges in distribution to N(0, 1).
In fact,

\sigma^2 = E(X_1^2) + 2 \sum_{k=1}^{\infty} E(X_1 X_{1+k}),
where the series converges absolutely.
The assumption σ ≠ 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:^{[9]} the assumption E(X_{n}^{12}) < ∞ is replaced with E(X_{n}^{2 + δ}) < ∞, and the assumption α_{n} = O(n^{−5}) is replaced with \sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty. Existence of such δ > 0 ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2005).
Martingale difference CLT
Theorem. Let a martingale M_{n} satisfy

\frac1n \sum_{k=1}^n \mathrm{E} ((M_kM_{k1})^2  M_1,\dots,M_{k1}) \to 1 in probability as n tends to infinity,

for every ε > 0, \frac1n \sum_{k=1}^n \mathrm{E} \Big( (M_kM_{k1})^2; M_kM_{k1} > \varepsilon \sqrt n \Big) \to 0 as n tends to infinity,
then M_n / \sqrt n converges in distribution to N(0,1) as n → ∞.^{[10]}^{[11]}
Caution: The restricted expectation E(X; A) should not be confused with the conditional expectation E(XA) = E(X; A)/P(A).
Proof of classical CLT
For a theorem of such fundamental importance to statistics and applied probability, the central limit theorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (weak) 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,

\varphi_Y(t) = 1  {t^2 \over 2} + o(t^2), \quad t \rightarrow 0
where o (t^{2}) is "little o notation" for some function of t that goes to zero more rapidly than t^{2}.
Letting Y_{i} be (X_{i} − μ)/σ, the standardized value of X_{i}, it is easy to see that the standardized mean of the observations X_{1}, X_{2}, ..., X_{n} is

Z_n = \frac{n\overline{X}_nn\mu}{\sigma \sqrt{n}} =\sum_{i=1}^n {Y_i \over \sqrt{n}}
By simple properties of characteristic functions, the characteristic function of the sum is:

\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)\cdots \varphi_{Y_n} \left(t / \sqrt{n} \right) = \left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n
so that, by the limit of the exponential function ( e^{x}= lim(1+x/n)^{n}) the characteristic function of Z_{n} is

\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.
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.
Convergence to the limit
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
If the third central moment E((X_{1} − μ)^{3}) exists and is finite, then the above convergence is uniform and the speed of convergence is at least on the order of 1/n^{1/2} (see BerryEsseen theorem). Stein's method^{[12]} can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.^{[13]}
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.^{[14]}
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 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.
Relation to the law of large numbers
The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behaviour of S_{n} as 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 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \qquad (n \rightarrow \infty).
Dividing both parts by φ_{1}(n) and taking the limit will produce a_{1}, the coefficient of the highestorder term in the expansion, which represents the rate at which f(n) changes in its leading term.

\lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.
Informally, one can say: "f(n) grows approximately as a_{1} φ_{1}(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):

\lim_{n\to\infty}\frac{f(n)a_1 \varphi_{1}(n)}{\varphi_{2}(n)}=a_2 .
Here one can say that the difference between the function and its approximation grows approximately as a_{2} φ_{2}(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 is happening when the sum, S_{n}, of independent identically distributed random variables, X_{1}, ..., X_{n}, is studied in classical probability theory. If each X_{i} has finite mean μ, then by the law of large numbers, S_{n}/n → μ.^{[15]} If in addition each X_{i} has finite variance σ^{2}, then by the central limit theorem,

\frac{S_nn\mu}{\sqrt{n}} \rightarrow \xi ,
where ξ is distributed as N(0, σ^{2}). This provides values of the first two constants in the informal expansion

S_n \approx \mu n+\xi \sqrt{n}. \,
In the case where the X_{i}'s do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:

\frac{S_na_n}{b_n} \rightarrow \Xi,
or informally

S_n \approx a_n+\Xi b_n. \,
Distributions Ξ which can arise in this way are called stable.^{[16]} 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_{n} may be proportional to n^{c}, for any c ≥ 1/2; it may also be multiplied by a slowly varying function of n.^{[17]}^{[18]}
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 \sqrt{n\log\log n} intermediate in size between n of the law of large numbers and √n of the central limit theorem provides a nontrivial limiting behavior.
Alternative statements of the theorem
Density functions
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,^{[19]} Chapter 7 for a particular local limit theorem for sums of i.i.d. random variables.
Characteristic functions
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. However, to state this more precisely, 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.
Extensions to the theorem
Products of positive random variables
The logarithm of a product is simply the sum of the logarithms of the factors. Therefore when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a lognormal 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 lognormal distribution.
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 squareintegrable.^{[20]}
Beyond the classical framework
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.
Convex body
Theorem. There exists a sequence ε_{n} ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X_{1}, ..., X_{n} have a logconcave joint density f such that f(x_{1}, ..., x_{n}) = f(x_{1}, ..., x_{n}) for all x_{1}, ..., x_{n}, and E(X_{k}^{2}) = 1 for all k = 1, ..., n. Then the distribution of

\frac{X_1+\cdots+X_n}{\sqrt n}
is ε_{n}close to N(0, 1) in the total variation distance.^{[21]}
These two ε_{n}close distributions have densities (in fact, logconcave 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 logconcave 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: f(x_{1}, …, x_{n}) = const · exp( − (x_{1}^{α} + … + x_{n}^{α})^{β}) where α > 1 and αβ > 1. If β = 1 then f(x_{1}, …, x_{n}) factorizes into const · exp ( − x_{1}^{α})…exp( − x_{n}^{α}), which means independence of X_{1}, …, X_{n}. In general, however, they are dependent.
The condition f(x_{1}, …, x_{n}) = f(x_{1}, …, x_{n}) ensures that X_{1}, …, X_{n} are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem.^{[22]}
Here is a BerryEsseen type result.
Theorem. Let X_{1}, …, X_{n} satisfy the assumptions of the previous theorem, then ^{[23]}

\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
for all a < b; here C is a universal (absolute) constant. Moreover, for every c_{1}, …, c_{n} ∈ R such that c_{1}^{2} + … + c_{n}^{2} = 1,

\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 ).
The distribution of (X_1+\dots+X_n)/\sqrt n need not be approximately normal (in fact, it can be uniform).^{[24]} However, the distribution of c_{1}X_{1} + … + c_{n}X_{n} is close to N(0, 1) (in the total variation distance) for most of vectors (c_{1}, …, c_{n}) according to the uniform distribution on the sphere c_{1}^{2} + … + c_{n}^{2} = 1.
Lacunary trigonometric series
Theorem (Salem–Zygmund). Let U be a random variable distributed uniformly on (0, 2π), and X_{k} = r_{k} cos(n_{k}U + a_{k}), where

n_{k} satisfy the lacunarity condition: there exists q > 1 such that n_{k+1} ≥ qn_{k} for all k,

r_{k} are such that


r_1^2 + r_2^2 + \cdots = \infty \text{ and } \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0,
Then^{[25]}^{[26]}

\frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} }
converges in distribution to N(0, 1/2).
Gaussian polytopes
Theorem Let A_{1}, ..., A_{n} be independent random points on the plane R^{2} each having the twodimensional standard normal distribution. Let K_{n} be the convex hull of these points, and X_{n} the area of K_{n} Then^{[27]}

\frac{ X_n  \mathrm{E} X_n }{ \sqrt{\operatorname{Var} X_n} }
converges in distribution to N(0, 1) as n tends to infinity.
The same holds in all dimensions (2, 3, ...).
The polytope K_{n} is called 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.^{[28]}
Linear functions of orthogonal matrices
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.
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.
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^{[29]} the distribution of X is close to N(0, 1) in the total variation metric up to 2√3/(n−1).
Subsequences
Theorem. Let random variables X_{1}, X_{2}, … ∈ L_{2}(Ω) be such that X_{n} → 0 weakly in L_{2}(Ω) and X_{n}^{2} → 1 weakly in L_{1}(Ω). Then there exist integers n_{1} < n_{2} < … such that ( X_{n_1}+\cdots+X_{n_k} ) / \sqrt k converges in distribution to N(0, 1) as k tends to infinity.^{[30]}
Tsallis statistics
A generalization of the classical central limit theorem to the context of Tsallis statistics has been described by Umarov, Tsallis and Steinberg^{[31]} 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 qGaussian distribution, which maximizes the Tsallis entropy under these constraints. Umarov, Tsallis, GellMann and Steinberg have defined similar generalizations of all symmetric alphastable distributions, and have formulated a number of conjectures regarding their relevance to an even more general Central limit theorem.^{[32]}
Random walk on a crystal lattice
The central limit theorem may be established for the simple random walk on a crystal lattice (an infinitefold abelian covering graph over a finite graph), and is used for design of crystal structures. ^{[33]}^{[34]}
Applications and examples
Simple example
Comparison of probability density functions, p(k) for the sum of n fair 6sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottomright graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
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 realworld 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 largesample statistics to the normal distribution in controlled experiments.
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
chisquared values that quantify the goodness of the fit (the fit is good if the reduced
chisquared 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/
√n), which is called the standard deviation of the mean (since it refers to the spread of sample means).
Real applications
A histogram plot of monthly accidental deaths in the US, between 1973 and 1978 exhibits normality, due to the central limit theorem
Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem.^{[35]} One source^{[36]} states the following examples:

The probability distribution for total distance covered in a random walk (biased or unbiased) will tend toward a normal distribution.

Flipping a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).
From another viewpoint, the central limit theorem explains the common appearance of the "Bell Curve" in 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.
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.
Regression
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 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.
Other illustrations
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.^{[37]}
History
Tijms writes:^{[38]}
The central limit theorem has an interesting history. The first version of this theorem was postulated by the Frenchborn mathematician PierreSimon 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 as:^{[39]}
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 selfeffacement, 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.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by

Hazewinkel, Michiel, ed. (2001), "Central limit theorem",

Animated examples of the CLT

Central Limit Theorem interactive simulation to experiment with various parameters

CLT in NetLogo (Connected Probability — ProbLab) interactive simulation w/ a variety of modifiable parameters

General Central Limit Theorem Activity & corresponding SOCR CLT Applet (Select the Sampling Distribution CLT Experiment from the dropdown list of SOCR Experiments)

Generate sampling distributions in Excel Specify arbitrary population, sample size, and sample statistic.

MIT OpenCourseWare Lecture 18.440 Probability and Random Variables, Spring 2011, Scott Sheffield Another proof. Retrieved 20120408.

CAUSEweb.org is a site with many resources for teaching statistics including the Central Limit Theorem

The Central Limit Theorem by Chris Boucher, Wolfram Demonstrations Project.

Weisstein, Eric W., "Central Limit Theorem", MathWorld.

Animations for the Central Limit Theorem by Yihui Xie using the R package animation

Teaching demonstrations of the CLT: clt.examp function in Greg Snow (2012). TeachingDemos: Demonstrations for teaching and learning. R package version 2.8..
External links


Bauer, Heinz (2001), Measure and Integration Theory, Berlin: de Gruyter,

Billingsley, Patrick (1995), Probability and Measure (Third ed.), John Wiley & sons,

Bradley, Richard (2007), Introduction to Strong Mixing Conditions (First ed.), Heber City, UT: Kendrick Press,

Bradley, Richard (2005), "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions", Probability Surveys 2: 107–144,

Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008), "Central Limit Theorem: New SOCR Applet and Demonstration Activity", Journal of Statistics Education (ASA) 16 (2)


Gaposhkin, V.F. (1966), "Lacunary series and independent functions", Russian Mathematical Surveys 21 (6): 1–82, .

Klartag, Bo'az (2007), "A central limit theorem for convex sets", Inventiones Mathematicae 168, 91–131.doi:10.1007/s0022200600288 Also arXiv.

Klartag, Bo'az (2008), "A BerryEsseen type inequality for convex bodies with an unconditional basis", Probability Theory and Related Fields. doi:10.1007/s0044000801586 Also arXiv.
References

^ http://www.math.uah.edu/stat/sample/CLT.html

^ Rice, John (1995), Mathematical Statistics and Data Analysis (Second ed.), Duxbury Press, )

^ Voit, Johannes (2003), The Statistical Mechanics of Financial Markets, SpringerVerlag, p. 124,

^ Billingsley (1995, p.357)

^ Bauer (2001, Theorem 30.13, p.199)

^ Billingsley (1995, p.362)

^ Van der Vaart, A. W. (1998), Asymptotic statistics, New York: Cambridge University Press,

^ Billingsley (1995, Theorem 27.4)

^ Durrett (2004, Sect. 7.7(c), Theorem 7.8)

^ Durrett (2004, Sect. 7.7, Theorem 7.4)

^ Billingsley (1995, Theorem 35.12)

^

^ Chen, L.H.Y., Goldstein, L., and Shao, Q.M (2011), Normal approximation by Stein's method, Springer,

^ Artstein, S.; Ball, K.; Barthe, F.;

^ Rosenthal, Jeffrey Seth (2000) A first look at rigorous probability theory, World Scientific, ISBN 9810243227.(Theorem 5.3.4, p. 47)

^ Johnson, Oliver Thomas (2004) Information theory and the central limit theorem, Imperial College Press, 2004, ISBN 1860944736. (p. 88)

^ Vladimir V. Uchaikin and V. M. Zolotarev (1999) Chance and stability: stable distributions and their applications, VSP. ISBN 9067643017.(pp. 61–62)

^ Borodin, A. N. ; Ibragimov, Il'dar Abdulovich; Sudakov, V. N. (1995) Limit theorems for functionals of random walks, AMS Bookstore, ISBN 0821804383. (Theorem 1.1, p. 8 )

^ Petrov, V.V. (1976), Sums of Independent Random Variables, New YorkHeidelberg: SpringerVerlag

^ Rempala, G.; Wesolowski, J.(2002) statistics"U"Asymptotics of products of sums and , Electronic Communications in Probability, 7, 47–54.

^ Klartag (2007, Theorem 1.2)

^ Durrett (2004, Section 2.4, Example 4.5)

^ Klartag (2008, Theorem 1)

^ Klartag (2007, Theorem 1.1)

^ . (2003 combined volume I,II: ISBN 0521890535) (Sect. XVI.5, Theorem 55)

^ Gaposhkin (1966, Theorem 2.1.13)

^ Bárány & Vu (2007, Theorem 1.1)

^ Bárány & Vu (2007, Theorem 1.2)

^ Meckes, Elizabeth (2008), "Linear functions on the classical matrix groups", Transactions of the American Mathematical Society 360 (10): 5355–5366,

^ Gaposhkin (1966, Sect. 1.5)

^ Umarov, Sabir; Tsallis, Constantino and Steinberg, Stanly (2008), "On a qCentral Limit Theorem Consistent with Nonextensive Statistical Mechanics", Milan j. Math. (Birkhauser Verlag) 76: 307–328,

^ Umarov, Sabir; Tsallis, Constantino, GellMann, Murray and Steinberg, Stanly (2010), "Generalization of symmetric αstable Lévy distributions for q > 1", J Math Phys. (American Institute of Physics) 51 (3): 033502,

^ Kotani, M.; Sunada, T (2003), Spectral geometry of crystal lattices, Contemporary Math., 338, 271–305.

^ Sunada T. (2012), Topological Crystallography With a View Towards Discrete Geometric Analysis", Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer

^ Dinov, Christou & Sanchez (2008)

^ SOCR CLT Activity wiki

^ Marasinghe, M., Meeker, W., Cook, D. & Shin, T.S.(1994 August), "Using graphics and simulation to teach statistical concepts", Paper presented at the Annual meeting of the American Statistician Association, Toronto, Canada.

^ Henk, Tijms (2004), Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press, p. 169,

^ Galton F. (1889) Natural Inheritance , p. 66

^ ^{a} ^{b}

^ ^{a} ^{b} ^{c}

^ Hald, Andreas A History of Mathematical Statistics from 1750 to 1930, Ch.17.

^ Fischer, Hans (2011), A History of the Central Limit Theorem: From Classical to Modern Probability Theory, Sources and Studies in the History of Mathematics and Physical Sciences, New York: Springer, (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)

^ 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, MoscowLeningrad, 174 pp.

^ Hodges, Andrew (1983) Alan Turing: the enigma. London: Burnett Books., pp. 8788.

^ Zabell, S.L. (2005) Symmetry and its discontents: essays on the history of inductive probability, Cambridge University Press. ISBN 0521444705. (pp. 199 ff.)

^ Aldrich, John (2009) "England and Continental Probability in the InterWar Years", Electronic Journ@l for History of Probability and Statistics, vol. 5/2, Decembre 2009. (Section 3)

^ Jørgensen, Bent (1997). The theory of dispersion models. Chapman & Hall.
Notes
See also
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 never published.^{[45]}^{[46]}^{[47]}
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Pólya, Lindeberg, Lévy, and Cramér during the 1920s, are given by Hans Fischer.^{[43]} Le Cam describes a period around 1935.^{[41]} Bernstein^{[44]} 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.
The occurrence of the Gaussian probability density 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 wellknown, 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. [...]
in 1920 translates as follows.
^{[40]} by PólyaOn the central limit theorem of calculus of probability and the problem of moments The abstract of the paper ^{[41]} in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".central 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 ^{[41]}[40]
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