Английская Википедия:Central limit theorem

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

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

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920,Шаблон:Sfnp thereby serving as a bridge between classical and modern probability theory.

An elementary form of the theorem states the following. Let <math>X_1, X_2, \dots, X_n</math> denote a random sample of <math>n</math> independent observations from a population with overall expected value (average) <math>\mu</math> and finite variance <math>\sigma^2</math>, and let <math>\bar{X}_n</math>denote the sample mean of that sample (which is itself a random variable). Then the limit as <math>n\to\infty</math> of the distribution of <math>\frac{\bar{X}_n-\mu}{\sigma_{\bar{X}_n}},</math> where <math>\sigma_{\bar{X}_n}=\frac{\sigma}{\sqrt{n}},</math> is the standard normal distribution.

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

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

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

Independent sequences

Файл:IllustrationCentralTheorem.png

Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.

Classical CLT

Let <math>\{X_1, \ldots, X_n}\</math> be a sequence of i.i.d. random variables having a distribution with expected value given by <math>\mu</math> and finite variance given by <math>\sigma^2.</math> Suppose we are interested in the sample average <math display="block">\bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}.</math>

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

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

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

In the case <math>\sigma > 0,</math> convergence in distribution means that the cumulative distribution functions of <math>\sqrt{n}(\bar{X}_n - \mu)</math> converge pointwise to the cdf of the <math>\mathcal{N}(0, \sigma^2)</math> distribution: for every real number <math>z,</math> <math display="block">\lim_{n\to\infty} \mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] = \lim_{n\to\infty} \mathbb{P}\left[\frac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma } \le \frac{z}{\sigma}\right]= \Phi\left(\frac{z}{\sigma}\right) ,</math> where <math>\Phi(z)</math> is the standard normal cdf evaluated at <math>z.</math> The convergence is uniform in <math>z</math> in the sense that <math display="block">\lim_{n\to\infty}\;\sup_{z\in\R}\;\left|\mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] - \Phi\left(\frac{z}{\sigma}\right)\right| = 0~,</math> where <math>\sup</math> denotes the least upper bound (or supremum) of the set.Шаблон:Sfnp

Lyapunov CLT

The theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the central limit theorem the random variables <math display="inline">X_i</math> have to be independent, but not necessarily identically distributed. The theorem also requires that random variables <math display="inline">\left| X_i\right|</math> have moments of some order Шаблон:Nowrap and that the rate of growth of these moments is limited by the Lyapunov condition given below.

Шаблон:Math theorem \, \sum_{i=1}^{n} \operatorname E\left[\left|X_{i} - \mu_{i}\right|^{2+\delta}\right] = 0</math> is satisfied, then a sum of <math display="inline">\frac{X_i - \mu_i}{s_n}</math> converges in distribution to a standard normal random variable, as <math display="inline">n</math> goes to infinity: <math display="block">\frac{1}{s_n}\,\sum_{i=1}^{n} \left(X_i - \mu_i\right) \ \xrightarrow{d}\ \mathcal{N}(0,1) .</math>}}

In practice it is usually easiest to check Lyapunov's condition for Шаблон:Nowrap

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 <math display="inline">\varepsilon > 0</math> <math display="block"> \lim_{n \to \infty} \frac{1}{s_n^2}\sum_{i = 1}^{n} \operatorname E\left[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\left\{\left| X_i - \mu_i \right| > \varepsilon s_n \right\}} \right] = 0</math> where <math display="inline">\mathbf{1}_{\{\ldots\}}</math> is the indicator function. Then the distribution of the standardized sums <math display="block">\frac{1}{s_n}\sum_{i = 1}^n \left( X_i - \mu_i \right)</math> converges towards the standard normal distribution Шаблон:Nowrap

Multidimensional CLT

Proofs that use characteristic functions can be extended to cases where each individual <math display="inline">\mathbf{X}_i</math> is a random vector in Шаблон:Nowrap with mean vector <math display="inline">\boldsymbol\mu = \operatorname E[\mathbf{X}_i]</math> and covariance matrix <math display="inline">\mathbf{\Sigma}</math> (among the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done component-wise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.

Let <math display="block">\mathbf{X}_i = \begin{bmatrix} X_{i(1)} \\ \vdots \\ X_{i(k)} \end{bmatrix}</math> be the Шаблон:Mvar-vector. The bold in <math display="inline">\mathbf{X}_i</math> means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be <math display="block">\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</math> and the average is <math display="block"> \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}</math> and therefore <math display="block">\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left[ \mathbf{X}_i - \operatorname E \left( \mathbf{X}_i \right) \right] = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} ( \mathbf{X}_i - \boldsymbol\mu ) = \sqrt{n}\left(\overline{\mathbf{X}}_n - \boldsymbol\mu\right)~. </math>

The multivariate central limit theorem states that <math display="block">\sqrt{n}\left( \overline{\mathbf{X}}_n - \boldsymbol\mu \right) \,\xrightarrow{D}\ \mathcal{N}_k(0,\boldsymbol\Sigma)</math> where the covariance matrix <math>\boldsymbol{\Sigma}</math> is equal to <math display="block"> \boldsymbol\Sigma = \begin{bmatrix} {\operatorname{Var} \left (X_{1(1)} \right)} & \operatorname{Cov} \left (X_{1(1)},X_{1(2)} \right) & \operatorname{Cov} \left (X_{1(1)},X_{1(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1(1)},X_{1(k)} \right) \\ \operatorname{Cov} \left (X_{1(2)},X_{1(1)} \right) & \operatorname{Var} \left( X_{1(2)} \right) & \operatorname{Cov} \left(X_{1(2)},X_{1(3)} \right) & \cdots & \operatorname{Cov} \left(X_{1(2)},X_{1(k)} \right) \\ \operatorname{Cov}\left (X_{1(3)},X_{1(1)} \right) & \operatorname{Cov} \left (X_{1(3)},X_{1(2)} \right) & \operatorname{Var} \left (X_{1(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1(3)},X_{1(k)} \right) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \operatorname{Cov} \left (X_{1(k)},X_{1(1)} \right) & \operatorname{Cov} \left (X_{1(k)},X_{1(2)} \right) & \operatorname{Cov} \left (X_{1(k)},X_{1(3)} \right) & \cdots & \operatorname{Var} \left (X_{1(k)} \right) \\ \end{bmatrix}~.</math>

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

It is unknown whether the factor <math display="inline">d^{1/4}</math> is necessary.

The Generalized Central Limit Theorem

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. The first published complete proof of the GCLT was in 1937 by Paul Lévy in French. An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.

The statement of the GCLT is as follows:

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.

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 α-mixing) defined by <math display="inline">\alpha(n) \to 0</math> where <math display="inline">\alpha(n)</math> is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is:Шаблон:Sfnp

Шаблон:Math theorem</math> converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math>.}}

In fact, <math display="block">\sigma^2 = \operatorname E\left(X_1^2\right) + 2 \sum_{k=1}^{\infty} \operatorname E\left(X_1 X_{1+k}\right),</math> where the series converges absolutely.

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

There is a stronger version of the theorem:Шаблон:Sfnp the assumption <math display="inline">\operatorname E\left[X_n^{12}\right] < \infty</math> is replaced with Шаблон:Nowrap and the assumption <math display="inline">\alpha_n = O\left(n^{-5}\right) </math> is replaced with <math display="block">\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.</math>

Existence of such <math display="inline">\delta > 0</math> ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see Шаблон:Harv.

Martingale difference CLT

Шаблон:Main Шаблон:Math theorem</math> converges in distribution to <math display="inline">\mathcal{N}(0, 1)</math> as <math display="inline">n \to \infty</math>.Шаблон:SfnpШаблон:Sfnp}}

Remarks

Proof of classical CLT

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

Assume <math display="inline">\{X_1, \ldots, X_n, \ldots \}</math> are independent and identically distributed random variables, each with mean <math display="inline">\mu</math> and finite variance Шаблон:Nowrap The sum <math display="inline">X_1 + \cdots + X_n</math> has mean <math display="inline">n\mu</math> and variance Шаблон:Nowrap Consider the random variable <math display="block">Z_n = \frac{X_1+\cdots+X_n - n \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{X_i - \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{1}{\sqrt{n}} Y_i,</math> where in the last step we defined the new random variables Шаблон:Nowrap each with zero mean and unit variance Шаблон:Nowrap The characteristic function of <math display="inline">Z_n</math> is given by <math display="block">\varphi_{Z_n}\!(t) = \varphi_{\sum_{i=1}^n {\frac{1}{\sqrt{n}}Y_i}}\!(t) \ =\ \varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right) \varphi_{Y_2}\!\! \left(\frac{t}{\sqrt{n}}\right)\cdots \varphi_{Y_n}\!\! \left(\frac{t}{\sqrt{n}}\right) \ =\ \left[\varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right)\right]^n, </math> where in the last step we used the fact that all of the <math display="inline">Y_i</math> are identically distributed. The characteristic function of <math display="inline">Y_1</math> is, by Taylor's theorem, <math display="block">\varphi_{Y_1}\!\left(\frac{t}{\sqrt{n}}\right) = 1 - \frac{t^2}{2n} + o\!\left(\frac{t^2}{n}\right), \quad \left(\frac{t}{\sqrt{n}}\right) \to 0</math> where <math display="inline">o(t^2 / n)</math> is "[[Little-o notation|little Шаблон:Mvar notation]]" for some function of <math display="inline">t</math> that goes to zero more rapidly than Шаблон:Nowrap By the limit of the exponential function Шаблон:Nowrap the characteristic function of <math>Z_n</math> equals <math display="block">\varphi_{Z_n}(t) = \left(1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right) \right)^n \rightarrow e^{-\frac{1}{2} t^2}, \quad n \to \infty.</math>

All of the higher order terms vanish in the limit Шаблон:Nowrap The right hand side equals the characteristic function of a standard normal distribution <math display="inline">\mathcal{N}(0, 1)</math>, which implies through Lévy's continuity theorem that the distribution of <math display="inline">Z_n</math> will approach <math display="inline">\mathcal{N}(0,1)</math> as Шаблон:Nowrap Therefore, the sample average <math display="block">\bar{X}_n = \frac{X_1+\cdots+X_n}{n}</math> is such that <math display="block">\frac{\sqrt{n}}{\sigma}(\bar{X}_n - \mu) = Z_n</math> converges to the normal distribution Шаблон:Nowrap from which the central limit theorem follows.

Convergence to the limit

The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.Шаблон:Citation needed

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment <math display="inline">\operatorname{E}\left[(X_1 - \mu)^3\right]</math> exists and is finite, then the speed of convergence is at least on the order of <math display="inline">1 / \sqrt{n}</math> (see Berry–Esseen theorem). Stein's method can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.

The convergence to the normal distribution is monotonic, in the sense that the entropy of <math display="inline">Z_n</math> 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 Шаблон:Mvar 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 Шаблон:Mvar approaches infinity; this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.

Common misconceptions

Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks. 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, allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See Z-test for where the approximation holds.

Relation to the law of large numbers

The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of Шаблон:Math as Шаблон:Mvar approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of <math display="inline">f(n)</math>: <math display="block">f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O\big(\varphi_{3}(n)\big) \qquad (n \to \infty).</math>

Dividing both parts by Шаблон:Math and taking the limit will produce Шаблон:Math, the coefficient of the highest-order term in the expansion, which represents the rate at which Шаблон:Math changes in its leading term. <math display="block">\lim_{n\to\infty} \frac{f(n)}{\varphi_{1}(n)} = a_1.</math>

Informally, one can say: "Шаблон:Math grows approximately as Шаблон:Math". Taking the difference between Шаблон:Math and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about Шаблон:Math: <math display="block">\lim_{n\to\infty} \frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)} = a_2 .</math>

Here one can say that the difference between the function and its approximation grows approximately as Шаблон:Math. 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, Шаблон:Mvar, of independent identically distributed random variables, Шаблон:Math, is studied in classical probability theory.Шаблон:Citation needed If each Шаблон:Mvar has finite mean Шаблон:Mvar, then by the law of large numbers, Шаблон:Math. If in addition each Шаблон:Mvar has finite variance Шаблон:Math, then by the central limit theorem, <math display="block"> \frac{S_n-n\mu}{\sqrt{n}} \to \xi ,</math> where Шаблон:Mvar is distributed as Шаблон:Math. This provides values of the first two constants in the informal expansion <math display="block">S_n \approx \mu n+\xi \sqrt{n}. </math>

In the case where the Шаблон:Mvar do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors: <math display="block">\frac{S_n-a_n}{b_n} \rightarrow \Xi,</math> or informally <math display="block">S_n \approx a_n+\Xi b_n. </math>

Distributions Шаблон:Math which can arise in this way are called stable. 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 Шаблон:Mvar may be proportional to Шаблон:Mvar, for any Шаблон:Math; it may also be multiplied by a slowly varying function of Шаблон:Mvar.

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 Шаблон:Math, intermediate in size between Шаблон:Mvar of the law of large numbers and Шаблон:Math of the central limit theorem, provides a non-trivial 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 Petrov for a particular local limit theorem for sums of independent and identically distributed 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. 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.

Calculating the variance

Let Шаблон:Mvar be the sum of Шаблон:Mvar random variables. Many central limit theorems provide conditions such that Шаблон:Math converges in distribution to Шаблон:Math (the normal distribution with mean 0, variance 1) as Шаблон:Math. In some cases, it is possible to find a constant Шаблон:Math and function Шаблон:Mvar such that Шаблон:Math converges in distribution to Шаблон:Math as Шаблон:Math.

Extensions

Products of positive random variables

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

Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.

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

These two Шаблон:Mvar-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: Шаблон:Math where Шаблон:Math and Шаблон:Math. If Шаблон:Math then Шаблон:Math factorizes into Шаблон:Math which means Шаблон:Math are independent. In general, however, they are dependent.

The condition Шаблон:Math ensures that Шаблон:Math are of zero mean and uncorrelated;Шаблон:Citation needed still, they need not be independent, nor even pairwise independent.Шаблон:Citation needed By the way, pairwise independence cannot replace independence in the classical central limit theorem.Шаблон:Sfnp

Here is a Berry–Esseen type result.

Шаблон:Math theorem \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le \frac{C}{n} </math> for all Шаблон:Math; here Шаблон:Mvar is a universal (absolute) constant. Moreover, for every Шаблон:Math such that Шаблон:Math, <math display="block"> \left| \mathbb{P} \left( a \le c_1 X_1+\cdots+c_n X_n \le b \right) - \frac{1}{\sqrt{2\pi}} \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le C \left( c_1^4+\dots+c_n^4 \right). </math>}}

The distribution of Шаблон:Math need not be approximately normal (in fact, it can be uniform).Шаблон:Sfnp However, the distribution of Шаблон:Math is close to <math display="inline"> \mathcal{N}(0, 1)</math> (in the total variation distance) for most vectors Шаблон:Math according to the uniform distribution on the sphere Шаблон:Math.

Gaussian polytopes

The same also holds in all dimensions greater than 2.

The polytope Шаблон:Mvar 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.Шаблон:Sfnp

Linear functions of orthogonal matrices

A linear function of a matrix Шаблон:Math is a linear combination of its elements (with given coefficients), Шаблон:Math where Шаблон:Math 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 Шаблон:Math; see Rotation matrix#Uniform random rotation matrices.

Random walk on a crystal lattice

The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.

Applications and examples

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

Файл:Central Limit Theorem.png

Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 512. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.

Regression

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

Other illustrations

Шаблон:Main Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.

History

Dutch mathematician Henk Tijms writes:

Sir Francis Galton described the Central Limit Theorem in this way:

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper. 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.

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.Шаблон:Sfnp 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.

See also

• Asymptotic equipartition property
• Asymptotic distribution
• Bates distribution
• Benford's law – result of extension of CLT to product of random variables.
• Berry–Esseen theorem
• Central limit theorem for directional statistics – Central limit theorem applied to the case of directional statistics
• Delta method – to compute the limit distribution of a function of a random variable.
• Erdős–Kac theorem – connects the number of prime factors of an integer with the normal probability distribution
• Fisher–Tippett–Gnedenko theorem – limit theorem for extremum values (such as Шаблон:Math)
• 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

References

• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal.
• Шаблон:Cite journal
• Шаблон:Cite journal

External links

• Central Limit Theorem at Khan Academy
• Шаблон:Springer
• Шаблон:MathWorld
• A music video demonstrating the central limit theorem with a Galton board by Carl McTague