Central limit theorem

In probability theory, the central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed.^[1] 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 non-identical distributions, given that they comply with conditions.

A simple example of the central limit theorem is rolling a large number of identical, biased 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.

In more general probability theory, a central limit theorem is any of a set of weak-convergence theories. 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 1/|x|^{α + 1} where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.^[2]

A generalized q-analog of the classical central limit theorem has been described by Umarov, Tsallis and Steinberg^[3] 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 q-Gaussian distribution, which maximizes the Tsallis entropy under these constraints. Umarov, Tsallis, Gell-Mann and Steinberg have defined q-analogs of all symmetric alpha-stable distributions, and have formulated a number of conjectures regarding their relevance to an even more general Central limit theorem.^[4]

This article is concerned only with the classical (i.e., i.i.d. and finite variance) central limit theorem.

Contents 1 Central limit theorems for independent sequences 1.1 Classical CLT 1.2 Lyapunov CLT 1.3 Lindeberg CLT 1.4 Multidimensional CLT 2 Central limit theorems for dependent processes 2.1 CLT under weak dependence 2.2 Martingale difference CLT 3 Remarks 3.1 Proof of classical CLT 3.2 Convergence to the limit 3.3 Relation to the law of large numbers 3.4 Illustration 3.5 Alternative statements of the theorem 3.5.1 Density functions 3.5.2 Characteristic functions 4 Extensions to the theorem 4.1 Products of positive random variables 4.2 Multivariate central limit theorem 5 Beyond the classical framework 5.1 Convex body 5.2 Lacunary trigonometric series 5.3 Gaussian polytopes 5.4 Linear functions of orthogonal matrices 5.5 Subsequences 6 Applications and examples 6.1 Signal processing 7 History 8 See also 9 Notes 10 References 11 External links

Central limit theorems for independent sequences

Classical CLT

Let be a random sample of size —that is, a sequence of independent and identically distributed random variables with expected values and variances . Suppose we are interested in the behavior of the sample average of these random variables: . Then the central limit theorem asserts that as gets larger, the distribution of approximates normal with mean µ and variance 1/nσ². The true strength of the theorem is that S_n approaches normality regardless of the shapes of the distributions of individual X_i’s. Formally, the theorem can be stated as follows:

Lindeberg–Lévy CLT: suppose {X_i} is a sequence of iid random variables with E[X_i] = µ and Var[X_i] = σ². Then as n approaches infinity, the random variable √n(S_n − µ) converges in distribution to a normal N(0, σ²):^[5]

Convergence in distribution means that the cumulative distribution function of √n(S_n − µ) converges pointwise to the cdf of the N(0, σ²) distribution: for any real number z,

where Φ(x) is the standard normal cdf.

Lyapunov CLT

The theorem is named after a 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] let {X_i} be a sequence of independent random variables, each having a finite expected value μ_i and variance σ 2
i . Define s 2
n  = ∑n
i = 1 σ 2
i . If for some δ > 0, the Lyapunov’s condition

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:

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, we can replace the Lyapunov condition with the following weaker one (from Lindeberg in 1920). For every ε > 0

where 1_{…} is the indicator function. Then the distribution of the standardized sums converges towards the standard normal distribution N(0,1).

Multidimensional CLT

We can easily extend proofs using characteristic functions for cases where each individual is an independent and identically distributed random vector in , with mean vector and covariance matrix Σ (amongst the individual components of the vector). Now, if we take the summations of these vectors as being done componentwise, then the Multidimensional central limit theorem states that when scaled, these converge to a multivariate normal distribution.^[7]

Let

be the i-vector. The bold in means that it is a random vector, not a random (univariate) variable.

Then the sum of the random vectors will be

and the average will be

and therefore

.

The multivariate central limit theorem states that

where the covariance matrix is equal to

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 α-mixing) defined by α(n) → 0 where α(n) is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is given in (Billingsley 1995, Theorem 27.4):

Theorem. Suppose that X₁, X₂, … is stationary and α-mixing with α_n = O(n⁻⁵) and that E(X_n) = 0 and E(X_n¹²) < ∞. Denote S_n = X₁ + … + X_n, then the limit σ² = lim _nn ^{− 1}E(S_n²) exists, and if σ ≠ 0 then converges in distribution to N(0, 1).

In fact, σ² = E(X₁²) + 2∑_k=1^∞E(X₁X_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.

For the theorem in full strength see (Durrett 1996, Sect. 7.7(c), Theorem (7.8)); the assumption E(X_n¹²) < ∞ is replaced with E(|X_n|^{2 + δ}) < ∞, and the assumption α_n = O(n ^{− 5}) is replaced with 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

•   in probability as n tends to infinity,
• for every ε > 0,     as n tends to infinity,

then converges in distribution to N(0,1) as n tends to infinity.

See (Durrett 1996, Sect. 7.7, Theorem (7.4)) or (Billingsley 1995, Theorem 35.12).

Caution: The restricted expectation E(X; A) should not be confused with the conditional expectation E(X|A) = E(X; A)/P(A).

Remarks

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,

where o (t² ) is "little o notation" for some function of t  that goes to zero more rapidly than t². 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 X₁, X₂, ..., X_n is

By simple properties of characteristic functions, the characteristic function of Z_n is

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((X₁ − 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 Berry-Esseen theorem).

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, as proven in Artstein, Ball, Barthe and Naor (2004).

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 behavior 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 ƒ(n):

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

Informally, one can say: "ƒ(n) grows approximately as a₁ φ(n)". Taking the difference between ƒ(n) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about ƒ(n):

Here one can say that the difference between the function and its approximation grows approximately as a₂ φ₂(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, X₁, ..., 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.^[8] If in addition each X_i has finite variance σ², then by the central limit theorem,

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

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:

or informally

Distributions Ξ which can arise in this way are called stable.^[9] 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.^[10][11]

The law of the iterated logarithm tells us 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 n of the law of large numbers and √n of the central limit theorem provides a non-trivial limiting behavior.

Illustration

Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem. ^[12]

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, under the conditions stated above.

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 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.

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 (see Rempala 2002).

Multivariate central limit theorem

If the i.i.d. random variate is an m-dimensional vector, represented as where j=0,1,...,m and the mean vector is:

and the covariance matrix for the m components is

and if is the j-th component in the i-th sample, then the sample mean for N trials is

and the central limit theorem for the distribution of will be:

where is the multivariate normal distribution for N trials, "| |" specifies the determinant, and summation is assumed over products with repeated indices. For N=1, and the univariate central limit theorem is recovered.

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 (Klartag 2007, Theorem 1.2). There exists a sequence ε_n ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X₁, …, X_n have a log-concave joint density f such that ƒ(x₁, …, x_n) = ƒ(|x₁|, …, |x_n|) for all x₁, …, x_n, and E(X_k²) = 1 for all k = 1, …, n. Then the distribution of is ε_n-close to N(0, 1) in the total variation distance.

These two ε_n-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: ƒ(x₁, …, x_n) = const · exp ( − (|x₁|^α + … + |x_n|^α)^β) where α > 1 and αβ > 1. If β = 1 then ƒ(x₁, …, x_n) factorizes into const · exp ( − |x₁|^α)…exp ( − |x_n|^α), which means independence of X₁, …, X_n. In general, however, they are dependent.

The condition ƒ(x₁, …, x_n) = ƒ(|x₁|, …, |x_n|) ensures that X₁, …, 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 (Durrett 1996, Section 2.4, Example 4.5).

Here is a Berry-Esseen type result.

Theorem (Klartag 2008, Theorem 1). Let X₁, …, X_n satisfy the assumptions of the previous theorem, then

for all a < b; here C is a universal (absolute) constant. Moreover, for every c₁, …, c_n ∈ R such that c₁² + … + c_n² = 1,

A more general case is treated in (Klartag 2007, Theorem 1.1). The condition ƒ(x₁, …, x_n) = ƒ(|x₁|, …, |x_n|) is replaced with much weaker conditions: E(X_k) = 0, E(X_k²) = 1, E(X_kX_ℓ) = 0 for 1 ≤ k < ℓ ≤ n. The distribution of need not be approximately normal (in fact, it can be uniform). However, the distribution of c₁X₁ + … + c_nX_n is close to N(0,1) (in the total variation distance) for most of vectors (c₁, …, c_n) according to the uniform distribution on the sphere c₁² + … + c_n² = 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_kU + 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

• 0 ≤ a_k < 2π.

Then

converges in distribution to N(0, 1/2).

See (Zygmund 1959, Sect. XVI.5, Theorem (5-5)) or (Gaposhkin 1966, Theorem 2.1.13).

Gaussian polytopes

Theorem (Barany & Vu 2007, Theorem 1.1). Let A₁, ..., A_n be independent random points on the plane R² each having the two-dimensional standard normal distribution. Let K_n be the convex hull of these points, and X_n the area of K_n Then

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 (Barany & Vu 2007, Theorem 1.2).

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 (Meckes 2008). 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 the distribution of X is close to N(0,1) in the total variation metric up to

Subsequences

Theorem (Gaposhkin 1966, Sect. 1.5). Let random variables X₁, X₂, … ∈ L₂(Ω) be such that X_n → 0 weakly in L₂(Ω) and X_n² → 1 weakly in L₁(Ω). Then there exist integers n₁ < n₂ < … such that converges in distribution to N(0, 1) as k tends to infinity.

Applications and examples

There are a number of useful and interesting examples and applications arising from the central limit theorem (Dinov, Christou & Sanchez 2008). See e.g. [1], presented as part of the SOCR CLT Activity.

• 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.

Signal processing

Signals can be smoothed by applying a Gaussian filter, which is just the convolution of a signal with an appropriately scaled Gaussian function. Due to the central limit theorem this smoothing can be approximated by several filter steps that can be computed much faster, like the simple moving average.

The central limit theorem implies that to achieve a Gaussian of variance σ² n filters with windows of variances σ₁², …, σ_n² with σ² = σ₁² + ⋯ + σ_n² must be applied.

History

Tijms (2004, p. 169) writes:

“

The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie Analytique des Probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.

”

Sir Francis Galton (Natural Inheritance, 1889) described the Central Limit Theorem as:

“

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and 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 George Pólya in 1920 in the title of a paper.^[13][14] 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".^[14] The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya^[13] in 1920 translates as follows.

“

The occurrence of the Gaussian probability density e−x2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by Liapounoff. [...]

”

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald.^[15] 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.^[16][17] Le Cam describes a period around 1935.^[14] See Bernstein (1945) for 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 never published.^[18][19][20]

See also

• Delta method – to compute the limit distribution of a function of a random variable.
• Berry–Esseen theorem – error bounds on normal approximations based on the central limit theorem
• Chernoff bound – better error bounds on normal approximations for the tails for certain distributions
• Illustration of the central limit theorem
• Law of large numbers – weaker conclusion in the same context
• Theorem of de Moivre–Laplace
• Sample size – required sample size for applying the central limit theorem
• Fisher–Tippett–Gnedenko theorem – limit theorem for extremum values (such as max{X_n})
• Central limit theorem for directional statistics - Central limit theorem applied to the case of directional statistics
• Erdős-Kac theorem - connects the number of prime factors of an integer with the normal probability

distribution

Notes

References

• S. Artstein, K. Ball, F. Barthe and A. Naor (2004), "Solution of Shannon's Problem on the Monotonicity of Entropy", Journal of the American Mathematical Society 17, 975–982. Also author's site.
• Barany, Imre; Vu, Van (2007), "Central limit theorems for Gaussian polytopes", The Annals of Probability (Institute of Mathematical Statistics) 35 (4): 1593–1621, doi:10.1214/009117906000000791 . Also arXiv.
• S.N.Bernstein, 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, Moscow-Leningrad, 1945. 174 pp.
• Billingsley, Patrick (1995), Probability and Measure (Third ed.), John Wiley & sons, ISBN 0-471-00710-2 
• Bradley, Richard (2007), Introduction to Strong Mixing Conditions (First ed.), Heber City, UT: Kendrick Press, ISBN 097404279X 
• Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008), "Central Limit Theorem: New SOCR Applet and Demonstration Activity", Journal of Statistics Education (ASA) 16 (2) . Also at ASA/JSE.
• Durrett, Richard (1996), Probability: theory and examples (Second ed.) 
• Fischer, H. (2010) A History of the Central Limit Theorem: From Classical to Modern Probability Theory, Springer. ISBN 0387878564
• Gaposhkin, V.F. (1966), "Lacunary series and independent functions", Russian Math. Surveys 21 (6): 1–82, doi:10.1070/RM1966v021n06ABEH001196 .
• Klartag, Bo'az (2007), "A central limit theorem for convex sets", Inventiones Mathematicae 168, 91–131.doi:10.1007/s00222-006-0028-8 Also arXiv.
• Klartag, Bo'az (2008), "A Berry-Esseen type inequality for convex bodies with an unconditional basis", Probability Theory and Related Fields. doi:10.1007/s00440-008-0158-6 Also arXiv.
• Le Cam, Lucien (1986), "The central limit theorem around 1935", Statistical Science 1:1, 78–91.
• Meckes, Elizabeth (2008), "Linear functions on the classical matrix groups", Transactions of the American Mathematical Society 360 (10): 5355–5366, doi:10.1090/S0002-9947-08-04444-9 . Also arXiv.
• Rempala, G. and J. Wesolowski, (2002) "Asymptotics of products of sums and U-statistics", Electronic Communications in Probability, 7, 47–54.
• Rice, John (1995), Mathematical Statistics and Data Analysis (Second ed.), Duxbury Press, ISBN 0-534-20934-3 
• Tijms, Henk (2004) Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press. ISBN 0521540364
• Zygmund, Antoni (1959), Trigonometric series, Volume II, Cambridge . (2003 combined volume I,II: ISBN 0521890535)

External links

• 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 drop-down list of SOCR Experiments)
• Generate sampling distributions in Excel Specify arbitrary population, sample size, and sample statistic.
• [2] Another proof.
• 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" from MathWorld.
• Animations for the Central Limit Theorem by Yihui Xie using the R package animation