Central limit theorem

The central limit theorem (CLT) states that if the sum of independent identically distributed random variables has a finite variance, then it will be approximately normally distributed (i.e., following a Gaussian distribution, or bell-shaped curve). 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 and 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 high frequency occurrence 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:

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma^2>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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma^2 /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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n \to \infty} \mbox{P}(Z_n \le z) = \Phi(z)\,,

or,

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow\infty}\mbox{P}\left(\frac{\overline{X}_n-\mu}{\sigma/ \sqrt{n}}\leq z\right)=\Phi(z)\,,

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \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,

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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. N represents the size of the population. 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 also applies to sums of independent and identical discrete random variables, although in this case the convergence of the sum toward a normal distribution has singular properties: namely, a sum of discrete random variables is still a discrete random variable, so that we are confronted to a series of discrete random variables whose probability distribution converges towards a probability density function corresponding to a continuous variable (namely 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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):

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \ (n \rightarrow \infty).

dividing both parts by Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi_{1}(n)

and taking the limit will produce Failed to parse (Missing texvc executable; please see math/README to configure.):  a_{1} 
- the coefficient at the highest-order term in the expansion representing the rate at which Failed to parse (Missing texvc executable; please see math/README to configure.):  f(n) 
changes in its leading term. 

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.

Informally, one can say: "Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)

 grows approximately as Failed to parse (Missing texvc executable; please see math/README to configure.):  a_1 \varphi_{1}(n) 

". Taking the difference between Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)

and its approximation and then dividing by the next term in the expansion we arrive to a more refined statement about Failed to parse (Missing texvc executable; please see math/README to configure.):  f(n) 

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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, Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{S_n}{n} \rightarrow \mu

and by The Central Limit Theorem, Failed to parse (Missing texvc executable; please see math/README to configure.):  \frac{S_n-n\mu}{\sqrt{n}} \rightarrow \xi 

where Failed to parse (Missing texvc executable; please see math/README to configure.): \xi

is distributed as Failed to parse (Missing texvc executable; please see math/README to configure.):  N(0,\sigma^2) 
 which provide values of first two constants in informal expansion:

Failed to parse (Missing texvc executable; please see math/README to configure.): S_n \approx \mu n+\xi \sqrt{n}.

It could be shown^{[citation needed]} that if X₁, X₂, X₃, ... are i.i.d. and Failed to parse (Missing texvc executable; please see math/README to configure.): E(|X_1|^{\beta}) < \infty

for some Failed to parse (Missing texvc executable; please see math/README to configure.): 1 \le \beta <2 
then Failed to parse (Missing texvc executable; please see math/README to configure.):  \frac{S_n-n\mu}{n^{\frac{1}{\beta}}} \to 0 
hence Failed to parse (Missing texvc executable; please see math/README to configure.):  \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 Failed to parse (Missing texvc executable; please see math/README to configure.):  \sqrt{n\log\log n} 
intermediate in size between n of The Law of Large Numbers and Failed to parse (Missing texvc executable; please see math/README to configure.):  \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

See also Lyapunov's central limit theorem.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): s_n^2 = \sum_{i = 1}^n \sigma_i^2.

Assume that the third central moments

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n \to \infty} \frac{r_n}{s_n} = 0.

(This is the Lyapunov condition). We again consider the sum Failed to parse (Missing texvc executable; please see math/README to configure.): S_n=X_1+\cdots+X_n , its expected value is Failed to parse (Missing texvc executable; please see math/README to configure.): m_n = \sum_{i=1}^{n}\mu_i

and its standard deviation is Failed to parse (Missing texvc executable; please see math/README to configure.): s_n

, if we standardize it by setting

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_n = \frac{S_n - m_n}{s_n}

then the distribution of Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 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

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

• m-dependent central limit theorem,
• martingale central limit theorem,
• central limit theorem for mixing processes and the
• central limit theorem for convex bodies.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma^2

 Failed to parse (Missing texvc executable; please see math/README to configure.): n
filters with windows of variances Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_1^2,\dots,\sigma_n^2
with Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma^2 = \sigma_1^2+\dots+\sigma_n^2
must be applied.

See also

• Diversification (finance)
• Illustration of the central limit theorem

Notes

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.

External links

• Animated examples of the CLT
• Central Limit Theorem Java
• 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.
• [1] Another proof.
• CAUSEweb.org is a site with many resources for teaching statistics including the Central Limit Theoremaf:Sentrale limietstelling

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