Central Limit Theorem

The central limit theorem (CLT) states that the sum of a large number of independent and identically-distributed random variables will be approximately normally distributed (i.e., following a Gaussian distribution, or bell-shaped curve) if the random variables have a finite variance. Formally, a central limit theorem is any of a set of weak-convergence results in probability theory. They all express the fact that any sum of many independent identically distributed random variables will tend to be distributed according to a particular "attractor distribution".

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 condition, Lyapunov 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.”

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.^[1] 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.^[2] 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, ... be a set of n independent and identically distributed random variables having finite values of mean µ and variance σ² > 0. The central limit theorem states that as the sample size n increases, the distribution of the sample average approaches the normal distribution with a mean µ and variance σ²/n irrespective of the shape of the original distribution.

Let the sum of the random variables be S_n, given by

S_n = X₁ + ... + X_n. Then, defining

$Z_n = \frac{S_n - n \mu}{\sigma \sqrt{n}}\,,$

the distribution of Z_n converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution).^[3] 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,

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

where

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

is the sample mean. ^[3] ^[4]

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 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

$Z_n = \frac{n\overline{X}_n-n\mu}{\sigma\sqrt{n}} = \sum_{i=1}^n {Y_i \over \sqrt{n}}.$

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

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^½ (see Berry-Esséen theorem).

The convergence normal is monotonic, in the sense that the entropy of Z_n increases monotonically to that of the normal distribution, as proven by Artstein, Ball, Barthe and Naor^{[citation needed]}.

Pictures of a distribution being "smoothed out" by summation (showing original density of distribution and three subsequent summations, obtained by convolution of density functions):

(See Illustration of the central limit theorem for further details on these images.)

A graphical representation of the central limit theorem can be formed by plotting random means of a population. Consider A_n. A_n will represent the mean of a random sample and X_n represents a single random variable from the sample:

A_n = (X₁ + ... + X_n) / n. Derive A_n from 1 to whichever sample size.

A₁ = (X₁) / 1

A₂ = (X₁ + X₂)/ 2

A₃ = (X₁ + X₂ + X₃)/3

For the CLT, it is recommended to plot the means upwards to 30 points (sample size 30).If we standardize A_n by setting Z_n = (A_n − μ) / (σ / n^½), we obtain the same variable Z_n as above, and it approaches a standard normal distribution.

The Central Limit Theorem, as an approximation for a finite number of observations, 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 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 to 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 $\infty$ . 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 is one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of f(n):

$f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \ (n \rightarrow \infty).$

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 f(n) changes in its leading term.

$\lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.$

Informally, one can say: "f(n) grows approximately as $a_1 \varphi_{1}(n)$ ". Taking the difference between f(n) and its approximation and then dividing by the next term in the expansion we arrive to a more refined statement about f(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 $a_2 \varphi_{2}(n)$ " 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 S_n is being studied in classical probability theory. Under certain regularity conditions, by The Law of Large Numbers, $\frac{S_n}{n} \rightarrow \mu$ and by The Central Limit Theorem, $\frac{S_n-n\mu}{\sqrt{n}} \rightarrow \xi$ where ξ is distributed as N(0,σ²) which provide values of first two constants in informal expansion:

$S_n \approx \mu n+\xi \sqrt{n}.$

It could be shown^{[citation needed]} 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.

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 tends to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above.

An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.

Products of positive random variables

The central limit theorem tells us what to expect about the sum of independent random variables, but what about the product? Well, the logarithm of a product is simply the sum of the logs of the factors, so the log of a product of random variables that take only positive values tends to have a normal distribution, which makes the product itself have a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the product 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).

Lyapunov condition

Let X_n be a sequence of independent random variables defined on the same probability space. Assume that X_n has finite expected value μ_n and finite standard deviation σ_n. We define

$s_n^2 = \sum_{i = 1}^n \sigma_i^2.$

Assume that the third central moments

$r_n^3 = \sum_{i = 1}^n \mbox{E}\left({\left| X_i - \mu_i \right|}^3 \right)$

are finite for every n, and that

$\lim_{n \to \infty} \frac{r_n}{s_n} = 0.$

(This is the Lyapunov condition). We again consider the sum $S_n=X_1+\cdots+X_n$ , its expected value is $m_n = \sum_{i=1}^{n}\mu_i$ and its standard deviation is s_n, if we standardize it by setting

$Z_n = \frac{S_n - m_n}{s_n}$

then the distribution of Z_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

\varepsilon s_n \right) = 0" class="tex" v:shapes="_x0000_i1058" border="0" height="50" width="381">

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

Non-independent case

There are some theorems which treat the case of sums of non-independent variables, for instance:

Applications and examples

There are a number of useful and interesting examples arising from the central limit theorem. Below are brief outlines of two such examples and here are a large number of CLT applications, 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).

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 $\sigma^2 = \sigma_1^2+\dots+\sigma_n^2$ must be applied.

Notes

^ Andreas Hald, History of Mathematical Statistics from 1750 to 1930, Ch.17.
^ Hans Fischer: (1) "The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods"; (2) "The Central Limit Theorem in the Twenties".
^ ^a ^b For decades, large sample size was set as n > 29; however, research since 1990, has indicated larger samples, such as 100 or 250, might be needed if the population is skewed far from normal: the more skew, the larger the sample needed. The conditions might be rare, but critical when they occur: computer animations are used to illustrate the cases. The cutoff with n > 29 has allowed Student-t tables to format in limited pages; however, that sample size might be too small. See below "Using graphics and simulation.." by Marasinghe et al, and see "Identification of Misconceptions in the Central Limit Theorem and Related Concepts and Evaluation of Computer Media as a Remedial Tool" by Yu, Chong Ho and Dr. John T. Behrens, Arizona State University & Spencer Anthony, Univ. of Oklahoma, Annual Meeting of the American Educational Research Association, presented April 19, 1995, paper revised in Feb 12, 1997, webpage (accessed 2007-10-25): CWisdom-rtf.
^ Marasinghe, M., Meeker, W., Cook, D. & Shin, T.S.(1994 August), "Using graphics and simulation to teach statistical concepts", Paper presented at the Annual meeting of the American Statistician Association, Toronto, Canada.

References

Henk Tijms, Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press, 2004.
S. Artstein, K. Ball, F. Barthe and A. Naor, "Solution of Shannon's Problem on the Monotonicity of Entropy", Journal of the American Mathematical Society 17, 975-982 (2004).
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.
G. Rempala and J. Wesolowski, "Asymptotics of products of sums and U-statistics", Electronic Communications in Probability, vol. 7, pp. 47-54, 2002.