Central Limit Theorem

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

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, thereby serving as a bridge between classical and modern probability theory.

If $X_1, X_2, \dots, X_n, \dots$ are random samples drawn from a population with overall mean $\mu$ and finite variance and if $\bar_n$ is the sample mean of the first $n$ samples, then the limiting form of the distribution, with $\sigma_\bar=\sigma/\sqrt$, is a standard normal distribution.

For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average 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.). In variants, 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

Classical CLT

Let $\\$ be a sequence of random samples — that is, a sequence of i.i.d. random variables drawn from a distribution of expected value given by $\mu$ and finite variance given by Suppose we are interested in the sample average
$\bar_n \equiv \frac$
of the first $n$ samples.

By the law of large numbers, the sample averages converge almost surely (and therefore also converge in probability) to the expected value $\mu$ as

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number $\mu$ during this convergence. More precisely, it states that as $n$ gets larger, the distribution of the difference between the sample average $\bar_n$ and its limit when multiplied by the factor $\sqrt$ approximates the normal distribution with mean 0 and variance For large enough , the distribution of $\bar_n$ gets arbitrarily close to the normal distribution with mean $\mu$ and variance

The usefulness of the theorem is that the distribution of $\sqrt(\bar_n - \mu)$ approaches normality regardless of the shape of the distribution of the individual Formally, the theorem can be stated as follows:

In the case convergence in distribution means that the cumulative distribution functions of $\sqrt(\bar_n - \mu)$ converge pointwise to the cdf of the $\mathcal(0, \sigma^2)$ distribution: for every real
\lim_ \mathbb\left sqrt(\bar_n-\mu) \le z\right= \lim_ \mathbb\left frac \le \frac\right \Phi\left(\frac\right) ,
where $\Phi(z)$ is the standard normal cdf evaluated The convergence is uniform in $z$ in the sense that
\lim_\;\sup_\;\left, \mathbb\left sqrt(\bar_n-\mu) \le z\right- \Phi\left(\frac\right)\ = 0~,
where $\sup$ denotes the least upper bound (or supremum) of the set.

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 $\left, X_i\$ have moments of some order and that the rate of growth of these moments is limited by the Lyapunov condition given below.

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

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 $\varepsilon > 0$
\lim_ \frac\sum_^ \mathbb\left X_i - \mu_i)^2 \cdot \mathbf_ \right= 0
where $\mathbf_$ is the indicator function. Then the distribution of the standardized sums
$\frac\sum_^n \left( X_i - \mu_i \right)$
converges towards the standard normal distribution

Multidimensional CLT

Proofs that use characteristic functions can be extended to cases where each individual $\mathbf_i$ is a random vector in with mean vector \boldsymbol\mu = \mathbb mathbf_i/math> and covariance matrix $\mathbf$ (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
$\mathbf_i = \begin X_ \\ \vdots \\ X_ \end$
be the -vector. The bold in $\mathbf_i$ means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be
\begin X_ \\ \vdots \\ X_ \end + \begin X_ \\ \vdots \\ X_ \end + \cdots + \begin X_ \\ \vdots \\ X_ \end = \begin \sum_^ \left X_ \right \\ \vdots \\ \sum_^ \left X_ \right \end = \sum_^ \mathbf_i
and the average is
$\frac \sum_^ \mathbf_i= \frac\begin \sum_^ X_ \\ \vdots \\ \sum_^ X_ \end = \begin \bar X_ \\ \vdots \\ \bar X_ \end = \mathbf$
and therefore
\frac \sum_^ \left \mathbf_i - \mathbb \left( X_i \right) \right= \frac\sum_^ ( \mathbf_i - \boldsymbol\mu ) = \sqrt\left(\overline_n - \boldsymbol\mu\right)~.

The multivariate central limit theorem states that
$\sqrt\left( \overline_n - \boldsymbol\mu \right) \,\xrightarrow\ \mathcal_k(0,\boldsymbol\Sigma)$
where the covariance matrix $\boldsymbol$ is equal to
$\boldsymbol\Sigma = \begin & \operatorname \left (X_,X_ \right) & \operatorname \left (X_,X_ \right) & \cdots & \operatorname \left (X_,X_ \right) \\ \operatorname \left (X_,X_ \right) & \operatorname \left( X_ \right) & \operatorname \left(X_,X_ \right) & \cdots & \operatorname \left(X_,X_ \right) \\ \operatorname\left (X_,X_ \right) & \operatorname \left (X_,X_ \right) & \operatorname \left (X_ \right) & \cdots & \operatorname \left (X_,X_ \right) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \operatorname \left (X_,X_ \right) & \operatorname \left (X_,X_ \right) & \operatorname \left (X_,X_ \right) & \cdots & \operatorname \left (X_ \right) \\ \end~.$

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

It is unknown whether the factor $d^$ is necessary.

Generalized theorem

The central limit theorem states that the sum of a number of independent and identically distributed random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with a power-law tail ( Paretian tail) distributions decreasing as $^$ where $0 < \alpha < 2$ (and therefore having infinite variance) will tend to a stable distribution $f(x; \alpha, 0, c, 0)$ as the number of summands grows. If $\alpha > 2$ then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian 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 $\alpha(n) \to 0$ where $\alpha(n)$ is so-called strong mixing coefficient.

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

In fact,
$\sigma^2 = \mathbb\left(X_1^2\right) + 2 \sum_^ \mathbb\left(X_1 X_\right),$
where the series converges absolutely.

The assumption $\sigma \ne 0$ cannot be omitted, since the asymptotic normality fails for $X_n = Y_n - Y_$ where $Y_n$ are another stationary sequence.

There is a stronger version of the theorem: the assumption \mathbb\left \right< \infty is replaced with and the assumption $\alpha_n = O\left(n^\right)$ is replaced with
$\sum_n \alpha_n^ < \infty.$

Existence of such $\delta > 0$ ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see .

Martingale difference CLT

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 $\$ are independent and identically distributed random variables, each with mean $\mu$ and finite variance The sum $X_1 + \cdots + X_n$ has mean $n\mu$ and variance Consider the random variable
$Z_n = \frac = \sum_^n \frac = \sum_^n \frac Y_i,$
where in the last step we defined the new random variables each with zero mean and unit variance The characteristic function of $Z_n$ is given by
\varphi_\!(t) = \varphi_\!(t) \ =\ \varphi_\!\!\left(\frac\right) \varphi_\!\! \left(\frac\right)\cdots \varphi_\!\! \left(\frac\right) \ =\ \left varphi_\!\!\left(\frac\right)\rightn,

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,
$\varphi_\!\left(\frac\right) = 1 - \frac + o\!\left(\frac\right), \quad \left(\frac\right) \to 0$
where $o(t^2 / n)$ is " little notation" for some function of $t$ that goes to zero more rapidly than By the limit of the exponential function the characteristic function of $Z_n$ equals
$\varphi_(t) = \left(1 - \frac + o\left(\frac\right) \right)^n \rightarrow e^, \quad n \to \infty.$

All of the higher order terms vanish in the limit The right hand side equals the characteristic function of a standard normal distribution $N(0, 1)$, which implies through Lévy's continuity theorem that the distribution of $Z_n$ will approach $N(0,1)$ as Therefore, the sample average
$\bar_n = \frac$
is such that
$\frac(\bar_n - \mu)$
converges to the normal distribution 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.

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment \operatorname\left X_1 - \mu)^3\right/math> exists and is finite, then the speed of convergence is at least on the order of $1 / \sqrt$ (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 $Z_n$ 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 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 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 behavior of as 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 $f(n)$:
$f(n)= a_1 \varphi_(n)+a_2 \varphi_(n)+O\big(\varphi_(n)\big) \qquad (n \to \infty).$

Dividing both parts by and taking the limit will produce , the coefficient of the highest-order term in the expansion, which represents the rate at which changes in its leading term.
$\lim_ \frac = a_1.$

Informally, one can say: " grows approximately as ". Taking the difference between and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about :
$\lim_ \frac = a_2 .$

Here one can say that the difference between the function and its approximation grows approximately as . 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, , of independent identically distributed random variables, , is studied in classical probability theory. If each has finite mean , then by the law of large numbers, . If in addition each has finite variance , then by the central limit theorem,
$\frac \to \xi ,$
where is distributed as . This provides values of the first two constants in the informal expansion
$S_n \approx \mu n+\xi \sqrt.$

In the case where the do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
$\frac \rightarrow \Xi,$
or informally
$S_n \approx a_n+\Xi b_n.$

Distributions 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 may be proportional to , for any ; it may also be multiplied by a slowly varying function of .

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 , intermediate in size between of the law of large numbers and 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 be the sum of random variables. Many central limit theorems provide conditions such that converges in distribution to (the normal distribution with mean 0, variance 1) as . In some cases, it is possible to find a constant and function such that converges in distribution to as .

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 -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: where and . If then factorizes into which means are independent. In general, however, they are dependent.

The condition ensures that 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.

Here is a Berry–Esseen type result.

The distribution of need not be approximately normal (in fact, it can be uniform). However, the distribution of is close to (in the total variation distance) for most vectors according to the uniform distribution on the sphere .

Lacunary trigonometric series

Gaussian polytopes

The same also holds in all dimensions greater than 2.

The polytope 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.

Linear functions of orthogonal matrices

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

Subsequences

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.

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

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. 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 )
* 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

Notes

References

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External links

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

{{DEFAULTSORT:Central Limit Theorem

Probability theorems
Theorems in statistics
Articles containing proofs
Asymptotic theory (statistics)