# Central limit theorem

In probability theory, the central limit theorem (CLT) states conditions under which the sum of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (Rice 1995). More generally, a central limit theorem is any of a set of weak-convergence results in probability theory. They all express the fact that a sum of many independent random variables will tend to be distributed according to one of a small set of "attractor" (i.e. stable) distributions.

Since many real populations yield distributions with finite variance, this explains the prevalence of the normal probability distribution. For other generalizations for finite variance which do not require identical distribution, see Lindeberg's condition, Lyapunov's condition, Gnedenko and Kolmogorov states.

## 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. ”

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 historic 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. A period around 1935 is described in (Le Cam 1986). 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 C.L.T. in a general setting.

## Classical central limit theorem

The central limit theorem is also known as the second fundamental theorem of probability. (The Law of large numbers is the first.)

Let X1, X2, X3, ... Xn be a sequence of n independent and identically distributed (i.i.d) random variables each having finite values of expectation µ and variance σ2 > 0. The central limit theorem states that as the sample size n increases  , the distribution of the sample average of these random variables approaches the normal distribution with a mean µ and variance σ2 / n irrespective of the shape of the original distribution.

Let the sum of n random variables be Sn, given by

Sn = X1 + ... + Xn. Then, defining a new random variable $Z_n = \frac{S_n - n \mu}{\sigma \sqrt{n}}\,,$

the distribution of Zn converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution). This is often written as $\sqrt{n}\left(\overline{X}_n - \mu\right)\ \stackrel{D}{\rightarrow}\ N(0,\sigma^2)\,,$

where $\overline{X}_n=S_n/n=(X_1+\cdots+X_n)/n\,$

is the sample mean.

This means: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have $\lim_{n \to \infty} \mbox{P}(Z_n \le z) = \Phi(z)\,,$

or, $\lim_{n\rightarrow\infty}\mbox{P}\left(\frac{\overline{X}_n-\mu}{\sigma/ \sqrt{n}}\leq z\right)=\Phi(z).$

### Proof of the central limit theorem

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 unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem, $\varphi_Y(t) = 1 - {t^2 \over 2} + o(t^2), \quad t \rightarrow 0$

where o (t2 ) is "little o notation" for some function of t  that goes to zero more rapidly than t2. Letting Yi be (Xi − μ)/σ, the standardized value of Xi, it is easy to see that the standardized mean of the observations X1, X2, ..., Xn is $Z_n = \frac{n\overline{X}_n-n\mu}{\sigma / \sqrt{n}} = \sum_{i=1}^n {Y_i \over \sqrt{n}}.$

By simple properties of characteristic functions, the characteristic function of Zn is $\left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n = \left[ 1 - {t^2 \over 2n} + o\left({t^2 \over n}\right) \right]^n \, \rightarrow \, e^{-t^2/2}, \quad n \rightarrow \infty.$

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.

### Convergence to the limit

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

If the third central moment E((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/n½ (see Berry-Esséen theorem).

The convergence to the normal distribution is monotonic, in the sense that the entropy of Zn 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. 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 Sn as n approaches infinity?" In mathematical analysis, asymptotic series is one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of ƒ(n): $f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \qquad (n \rightarrow \infty).$

dividing both parts by $\varphi_{1}(n)$ and taking the limit will produce a1 — the coefficient at the highest-order term in the expansion representing the rate at which ƒ(n) changes in its leading term. $\lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.$

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 to a more refined statement about ƒ(n): $\lim_{n\to\infty}\frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)}=a_2$

here one can say that: "the difference between the function and its approximation grows approximately as 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 Sn is being studied in classical probability theory. Under certain regularity conditions, by The Law of Large Numbers, Sn → μ and by The Central Limit Theorem, $\frac{S_n-n\mu}{\sqrt{n}} \rightarrow \xi$

where ξ is distributed as N(0, σ2), which provides values of first two constants in informal expansion: $S_n \approx \mu n+\xi \sqrt{n}. \,$

It could be shown that if X1, X2, X3, ... are i.i.d. and $E(|X_1|^{\beta}) < \infty$ for some $1 \le \beta <2$ then $\frac{S_n-n\mu}{n^{\frac{1}{\beta}}} \to 0$ hence $\sqrt{n}$ is the largest power of n which if serves as a normalizing function would provide a non-trivial (non-zero) limiting behavior. Interestingly enough, 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 $\sqrt{n\log\log n}$ intermediate in size between n of The Law of Large Numbers and $\sqrt{n}$ 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, 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

### Multidimensional central limit theorem

We can easily extend proofs using characteristic functions for cases where each individual Xi is an independent and identically distributed random vector, with mean vector μ 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. $\sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right)\ \stackrel{D}{\rightarrow}\ \mathcal{N}(0,\Sigma)$

### 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).

### Lack of identical distribution

The central limit theorem also applies in the case of sequences that are not identically distributed, provided one of a number of conditions apply.

#### Lyapunov condition

Let Xn be a sequence of independent random variables defined on the same probability space. Assume that Xn has finite expected value μn and finite standard deviation σn. We define $s_n^2 = \sum_{i = 1}^n \sigma_i^2.$

If for some δ > 0, the expected values $\mathbb{E}[|X_{i}|^{2+\delta}]$ are finite for every $i \in \mathbb{N}$ and the Lyapunov's condition $\lim_{n\to\infty} \frac{1}{s_{n}^{2+\delta}} \sum_{i=1}^{n} \mathbb{E}[|X_{i} - \mu_{i}|^{2+\delta}] = 0$

is satisfied, then the distribution of the random variable $Z_{n} := \frac{\sum_{i = 1}^{n} \left( X_{i} - \mu_{i} \right)}{s_{n}}$

converges to the standard normal distribution N(0,1).

#### Lindeberg condition

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 $\lim_{n \to \infty} \sum_{i = 1}^{n} \mbox{E}\left( \frac{(X_i - \mu_i)^2}{s_n^2} : \left| X_i - \mu_i \right| > \varepsilon s_n \right) = 0$

where E( U : V > c) is the expectation of the random variable U |{V > c} whose value is U if V > c and zero otherwise. Then the distribution of the standardized sum Zn converges towards the standard normal distribution N(0,1).

## 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.

### Central limit theorem 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 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 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 $X_1, X_2, \dots$ is stationary and α-mixing with $\alpha_n = O(n^{-5}) \,$ and that $\mathrm{E}(X_n)=0 \,$ and $\mathrm{E}(X_n^{12}) < \infty$. Denote $S_n = X_1+\dots+X_n,$ then the limit $\sigma^2 = \lim_n n^{-1} \mathrm{E}(S_n^2)$ exists, and if $\sigma \ne 0$ then $S_n / (\sigma \sqrt n)$ converges in distribution to N(0,1).

In fact, $\sigma^2 = \mathrm{E}(X_1^2) + 2 \sum_{k=1}^\infty \mathrm{E}(X_1 X_{1+k}),$ where the series converges absolutely.

The assumption $\sigma \ne 0$ cannot be omitted, since the asymptotic normality fails for $X_n = Y_n - Y_{n-1} \,$ where Yn are another stationary sequence.

For the theorem in full strength see (Durrett 1996, Sect. 7.7(c), Theorem (7.8)); the assumption $\mathrm{E}(X_n^{12}) < \infty$ is replaced with $\mathrm{E}(|X_n|^{2+\delta}) < \infty,$ and the assumption αn = O(n − 5) is replaced with $\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.$ Existence of such δ > 0 ensures the conclusion.

### Martingale central limit theorem

Theorem. Let a martingale Mn satisfy

• $\frac1n \sum_{k=1}^n \mathrm{E} ((M_k-M_{k-1})^2 | M_1,\dots,M_{k-1}) \to 1$   in probability as n tends to infinity,
• for every $\varepsilon>0,$ $\frac1n \sum_{k=1}^n \mathrm{E} \Big( (M_k-M_{k-1})^2; |M_k-M_{k-1}| > \varepsilon \sqrt n \Big) \to 0$   as n tends to infinity,

then $M_n / \sqrt n$ converges in distribution to N(0,1) as n 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 $\mathrm{E}(X|A) = \mathrm{E}(X;A) / \mathbb{P}(A).$

### Central limit theorem for convex bodies

Theorem (Klartag 2007, Theorem 1.2). There exists a sequence $\varepsilon_n \downarrow 0$ for which the following holds. Let $n \ge 1$, and let random variables $X_1,\dots,X_n$ have a log-concave joint density f such that $f(x_1,\dots,x_n) = f(|x_1|,\dots,|x_n|)$ for all $x_1,\dots,x_n,$ and $\mathrm{E}(X_k^2) = 1$ for all $k = 1,\dots,n.$ Then the distribution of $(X_1+\dots+X_n)/\sqrt n$ is $\varepsilon_n$-close to N(0,1) in the total variation distance.

These two $\varepsilon_n$-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: $f(x_1,\dots,x_n) = \mathrm{const} \cdot \exp ( - ( |x_1|^\alpha + \dots + |x_n|^\alpha )^\beta )$ where α > 1 and αβ > 1. If β = 1 then $f(x_1,\dots,x_n)$ factorizes into $\mathrm{const} \cdot \exp (-|x_1|^\alpha) \dots \exp (-|x_n|^\alpha),$ which means independence of $X_1,\dots,X_n.$ In general, however, they are dependent.

The condition $f(x_1,\dots,x_n) = f(|x_1|,\dots,|x_n|)$ ensures that $X_1,\dots,X_n$ 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 $X_1,\dots,X_n$ satisfy the assumptions of the previous theorem, then $\bigg| \mathbb{P} \Big( a \le \frac{ X_1+\dots+X_n }{ \sqrt n } \le b \Big) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le \frac C n$

for all $a < b; \,$ here $C \,$ is a universal (absolute) constant. Moreover, for every $c_1,\dots,c_n \in \mathbb{R}$ such that $c_1^2+\dots+c_n^2 = 1,$ $\bigg| \mathbb{P} ( a \le c_1 X_1+\dots+c_n X_n \le b ) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le C ( c_1^4+\dots+c_n^4 ).$

A more general case is treated in (Klartag 2007, Theorem 1.1). The condition $f(x_1,\dots,x_n) = f(|x_1|,\dots,|x_n|)$ is replaced with much weaker conditions: E(Xk) = 0, $\mathrm{E}(X_k^2) = 1,$ E(XkXl) = 0 for $1 \le k < l \le n.$ The distribution of $(X_1+\dots+X_n)/\sqrt n$ need not be approximately normal (in fact, it can be uniform). However, the distribution of $c_1 X_1+\dots+c_n X_n$ is close to N(0,1) (in the total variation distance) for most of vectors $(c_1,\dots,c_n)$ according to the uniform distribution on the sphere $c_1^2+\dots+c_n^2 = 1.$

### Central limit theorem for lacunary trigonometric series

Theorem (Salem - Zygmund). Let U be a random variable distributed uniformly on (0, 2π), and Xk = rk cos(nkU + ak), where

• nk satisfy the lacunarity condition: there exists q > 1 such that nk+1 ≥ qnk for all k,
• rk are such that $r_1^2 + r_2^2 + \cdots = \infty \text{ and } \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0,$
• 0 ≤ ak < 2π.

Then $\frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} }$

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

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

### Central limit theorem for Gaussian polytopes

Theorem (Barany & Vu 2007, Theorem 1.1). Let A1, ..., An be independent random points on the plane R2 each having the two-dimensional standard normal distribution. Let Kn be the convex hull of these points, and Xn the area of Kn Then $\frac{ X_n - \mathrm{E} X_n }{ \sqrt{\operatorname{Var} X_n} }$

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

The same holds in all dimensions (2, 3, ...).

The polytope Kn 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).

### Central limit theorem for linear functions of orthogonal matrices

A linear function of a matrix M is a linear combination of its elements (with given coefficients), $M \mapsto \operatorname{tr} (AM)$ 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 $\operatorname{tr} (AA^*) = n,$ and let $X = \operatorname{tr} (AM).$ Then the distribution of X is close to N(0,1) in the total variation metric up to $2\sqrt3 / (n-1).$

### Central limit theorem for subsequences

Theorem (Gaposhkin 1966, Sect. 1.5). Let random variables $X_1,X_2,\dots \in L_2(\Omega)$ be such that $X_n \to 0$ weakly in L2(Ω) and $X_n^2 \to 1$ weakly in L1(Ω). Then there exist integers $n_1 < n_2 < \dots$ such that $( X_{n_1}+\cdots+X_{n_k} ) / \sqrt k$ 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. , 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 like the sum of independent variables with equal influence on the result a measurement is, 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 $\sigma_1^2,\dots,\sigma_n^2$ with $\sigma^2 = \sigma_1^2+\cdots+\sigma_n^2$ must be applied.