Central limit theorem

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 form it was only precisely stated in the 1920s.

In statistics, the CLT can be stated as: let <math>X_1, X_2, \dots, X_n</math> denote a statistical sample of size <math>n</math> from a population with expected value (average) <math>\mu</math> and finite positive variance <math>\sigma^2</math>, and let <math>\bar{X}_{n}</math> denote the sample mean (which is itself a random variable). Then the limit as <math>n\to\infty</math> of the distribution of <math>(\bar{X}_n-\mu) \sqrt{n} </math> is a normal distribution with mean <math> 0 </math> and variance <math>\sigma^2</math>.

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 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 is 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

thumb|upright=1.4 |right|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_n)_{n\geq 1}</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 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 normalized mean <math>\sqrt{n}(\bar{X}_n - \mu)</math>, i.e. the difference between the sample average <math>\bar{X}_n</math> and its limit <math>\mu,</math> scaled by the factor <math>\sqrt{n}</math>, approaches 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:

\mathcal{N}\left(0,\sigma^2\right) .</math>

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 supremum (i.e. least upper bound) of the set.

Lyapunov CLT

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 and that the rate of growth of these moments is limited by the Lyapunov condition given below.

\, \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) \mathrel{\overset{d}{\longrightarrow \mathcal{N}(0,1) .</math>

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 (-Feller) 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

CLT for the sum of a random number of random variables

Rather than summing an integer number <math>n</math> of random variables and taking <math>n \to \infty</math>, the sum can be of a random number <math>N</math> of random variables, with conditions on <math>N</math>. For example, the following theorem is Corollary 4 of Robbins (1948). It assumes that <math>N</math> is asymptotically normal (Robbins also developed other conditions that lead to the same result).

\xrightarrow{\quad d \quad} \mathcal{N}(0,1)

</math>

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

Then

\frac{\sum_{i=1}^{N_n} X_i - \mu E(N_n)}{\sqrt{\sigma^2E(N_n) + \mu^2\text{Var}(N_n) \xrightarrow{\quad d \quad} \mathcal{N}(0,1)

</math>

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 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. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution. Summation of these vectors is done component-wise.

For <math>i = 1, 2, 3, \ldots,</math> let

<math display="block">\mathbf{X}_i = \begin{bmatrix} X_{i}^{(1)} \\ \vdots \\ X_{i}^{(k)} \end{bmatrix}</math>

be independent random vectors. The sum of the random vectors <math>\mathbf{X}_1, \ldots, \mathbf{X}_n</math> is

<math display="block">\sum_{i=1}^{n} \mathbf{X}_i = \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} X_{i}^{(1)} \\ \vdots \\ \sum_{i=1}^{n} X_{i}^{(k)} \end{bmatrix}</math>

and their average is

<math display="block">\mathbf{\bar X_n} = \begin{bmatrix} \bar X_{i}^{(1)} \\ \vdots \\ \bar X_{i}^{(k)} \end{bmatrix} = \frac{1}{n} \sum_{i=1}^{n} \mathbf{X}_i.</math>

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) \mathrel{\overset{d}{\longrightarrow \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 multivariate central limit theorem can be proved using the Cramér–Wold theorem.

| math_statement = Let <math>X_1, \dots, X_n, \dots</math> be independent <math>\R^d</math>-valued random vectors, each having mean zero. Write <math>S =\sum^n_{i=1}X_i</math> and assume <math>\Sigma = \operatorname{Cov}[S]</math> is invertible. Let <math>Z \sim \mathcal{N}(0,\Sigma)</math> be a <math>d</math>-dimensional Gaussian with the same mean and same covariance matrix as <math>S</math>. Then for all convex sets

<math display="block">\left|\mathbb{P}[S \in U] - \mathbb{P}[Z \in U]\right| \le C \, d^{1/4} \gamma~,</math>

where <math>C</math> is a universal constant, and <math>\|\cdot\|_2</math> denotes the Euclidean norm on

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 (Sergei Bernstein, Jarl Waldemar Lindeberg, Paul Lévy, William Feller, Andrey 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 Boris Vladimirovich Gnedenko and Kolmogorov's 1954 book.

The statement of the GCLT is as follows:

In other words, if sums of independent, identically distributed random variables converge in distribution to some , then 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:

</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: the assumption <math display="inline">\operatorname E\left[X_n^{12}\right] < \infty</math> is replaced with 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 .

Martingale difference CLT

</math> converges in distribution to <math display="inline">\mathcal{N}(0, 1)</math> as <math display="inline">n \to \infty</math>.

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 The sum <math display="inline">X_1 + \cdots + X_n</math> has mean <math display="inline">n\mu</math> and variance 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 the last step defines the new random variables each with zero mean and unit variance The characteristic function of <math display="inline">Z_n</math> is given by

<math display="block">\begin{align}

\varphi_{Z_n}\!(t) = \varphi_{\sum_{i=1}^n {\frac{1}{\sqrt{nY_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) \\[1ex]

&=\ \left[\varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n\right)\right]^n,

\end{align}

</math>

where the last step relies on 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 notation" for some function of <math display="inline">t</math> that goes to zero more rapidly than By the limit of the exponential function 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 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 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} \left(\bar{X}_n - \mu\right) = Z_n</math>

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 <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. 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 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 <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 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.

<math display="block">\lim_{n\to\infty} \frac{f(n)}{\varphi_{1}(n)} = a_1.</math>

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 :

<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 . 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,

where is distributed as . 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 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 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.

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.

center|thumb|820px|Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. 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

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:

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

Notes

References

External links

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

Independent sequences

Classical CLT

Lyapunov CLT

Lindeberg (-Feller) CLT

CLT for the sum of a random number of random variables

Multidimensional CLT

The generalized central limit theorem

Dependent processes

CLT under weak dependence

Martingale difference CLT

Remarks

Proof of classical CLT

Convergence to the limit

Relation to the law of large numbers

Alternative statements of the theorem

Density functions

Characteristic functions

Calculating the variance

Extensions

Products of positive random variables

Applications and examples

Regression

Other illustrations

History

See also

Notes

References

External links