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1.14: 14 Central limit theorems - Mathematics


1.14: 14 Central limit theorems - Mathematics

Central Limit Theorem

The central limit theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend towards the normal distribution as the sample size gets larger. This may be restated as follows:

Given a set of independent and identically distributed random variables
X1, X2, . Xn, where E(Xi) = m and
Var(Xi) = s 2 , i = 1, 2, . n, and the mean of the random variables given by

the distribution of approaches the normal distribution, i.e.

The central limit theorem is one of the most important theorems in the field of probability as well as statistical inference, as it justifies the use of the normal curve in a wide range of statistical applications, both theoretical and practical. The approximate normality of the sampling distribution of the mean is usually achieved when n ³ 30 (except in certain cases where the probability distribution of the population has a very unusual shape). A Standard Normal Distribution Table can then be used in these situations to solve for probabilities involving the sample mean.

The normal approximation to the binomial distribution is a special case of the central limit theorem, where the independent random variables are Bernoulli variables with parameter p.


1.14: 14 Central limit theorems - Mathematics

Laboratoire de Mathématiques Raphaël Salem
Unité Mixte de Recherche 6085 CNRS - Université de Rouen

Université de Rouen - Technopôle du Madrillet

Faculté des Sciences et Techniques

Avenue de l'Université - BP 12

Mon travail de recherche est centré sur l'étude de théorèmes limites pour les variables aléatoires dépendantes et leurs applications en statistique.


Travaux publiés ou soumis pour publication dans des revues avec comité de lecture :

16. On local linear regression for strongly mixing random fields. [pdf]
M. El Machkouri, K. Es-Sebaiy and I. Ouassou.
To appear in Journal of Multivariate Analysis.

15. Parameter estimation for the non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian process. [pdf]
M. El Machkouri, K. Es-Sebaiy and Y. Ouknine.
To appear in Journal of the Korean Statistical Society.

14. Orthomartingale-coboundary decomposition for stationary random fields. [pdf]
M. El Machkouri and D. Giraudo.
To appear in Stochastics and Dynamics.

13. Kernel density estimation for stationary random fields [pdf]
M. El Machkouri.
ALEA, Lat. Am. J. Probab. Math. Stat. , 259-279, 16 , N o 11 (1), 2014.

12. On the asymptotic normality of frequency polygons for random fields [pdf]
M. El Machkouri.
Statistical Inference for Stochastic Processes , 193-206, 16 , N o 3, 2013.

11. A central limit theorem for stationary random fields [pdf]
M. El Machkouri, D. Volný and W. B. Wu.
Stochastic Processes and Their Applications , 1-14, 123 , 2013.

10. Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields [pdf]
M. El Machkouri.
Statistical Inference for Stochastic Processes , 73-84, 14 , N o 1, 2011.

9. Asymptotic normality of kernel estimates in a regression model for random fields [pdf]
M. El Machkouri and R. Stoica.
Journal of Nonparametric Statistics , 955-971, 22 , N o 8, 2010.

8. A criterion of weak mixing property [pdf]
E.H. El Abdalaoui, M. El Machkouri and A. Nogueira
Séminaires et congrès de la SMF , 105-111, 20 , 2010.

7. Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space [pdf]
M. El Machkouri
African Diaspora Journal of Mathematics , 81-86, 10 , N o 2, 2010.

6. Exact convergence rates in the central limit theorem for a class of martingales [pdf]
M. El Machkouri and L. Ouchti.
Bernoulli , 981-999, 13 , N o 4, 2007.

5. Nonparametric regression estimation for random fields in a fixed-design [pdf]
M. El Machkouri.
Statistical Inference for Stochastic Processes , 29-47, 10 , N o 1, 2007.

4. Invariance principles for standard-normalized and self-normalized random fields [pdf]
M. El Machkouri and L. Ouchti.
ALEA , 177-194, 2 , 2006.

3. On the local and central limit theorems for martingale difference sequences [pdf]
M. El Machkouri and D. Volný.
Stochastics and Dynamics , 1-21, 4 , N o 2, 2004.

2. Contre-exemple dans le théorème limite central fonctionnel pour les champs aléatoires réels [pdf]
M. El Machkouri et D. Volný.
Annales de l'Institut Henri Poincaré (B) Probability and Statistics , 325-337, 39 , N o 2, 2003.

1. Kahane-Khintchine inequalities and functional central limit theorem for stationary real random fields [pdf]
M. El Machkouri.
Stochastic Processes and Their Applications , 285-299, 120 , 2002.


Publications et prépublications

Cônes limites des sous-groupes discrets des groupes réductifs sur un corps local, Transformation groups 7 (2002), 247-266 (pdf).

Divergence exponentielle des sous-groupes discrets en rang supérieur, Commentarii mathematici helvetici 77 (2002), 563-608 (pdf).

Mesures de Patterson-Sullivan en rang supérieur, Geometric and functional analysis 12 (2002), 776-809 (pdf).

L'indicateur de croissance des groupes de Schottky, Ergodic theory and dynamical systems 23 (2003), 249-272 (pdf).

Propriété de Kazhdan et sous-groupes discrets de covolume infini, Travaux mathématiques 14 (2003), 143-151 (pdf).

Groupes de Schottky et comptage, Annales de l'Institut Fourier 55 (2005), 373-429 (pdf).

Groupes convexes cocompacts en rang supérieur, Geometriae dedicata 113 (2005), 1-19 (pdf).

Avec H. Dathe, Exemples de feuilletages de Lie, Annales de la Faculté des Sciences de Toulouse 15 (2006), 203-215 (pdf).

Harmonic analysis on the Pascal graph, Journal of functional analysis 256 (2009), 3409-3460 (pdf).

Percolation on the three dot system, Probability theory and related fields 146 (2010), 49-60 (pdf).

Avec Y. Benoist, Mesures stationnaires et fermés invariants des espaces homogènes, Annals of mathematics 174 (2011), 1111-1162 (pdf).

Avec Y. Benoist, Random walks on finite volume homogeneous spaces, Inventiones mathematicae 187 (2012), 37-59 (pdf).

Avec Y. Benoist, Stationary measures and invariant subsets of homogeneous spaces (II), Journal of the American Mathematical Society 26 (2013), 659-734. (pdf).

Avec Y. Benoist, Stationary measures and invariant subsets of homogeneous spaces (III), Annals of mathematics 178 (2013), 1017-1059 (pdf).

Avec Y. Benoist, Lattices in S-adic Lie groups, Journal of Lie theory 24 (2014), 179-197 (pdf).

Avec Y. Benoist, Random walks on projective spaces, Compositio mathematica 150 (2014), 1579-1606 (pdf).

Avec Y. Benoist, Central limit theorem for linear groups, Annals of probability 44 (2016), 1308-1340 (pdf).

Avec Y. Benoist, Central limit theorem for hyperbolic groups, Izvestiya mathematics 80 (2016), 5-26 (pdf).

Avec Y. Benoist, Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 62, Springer, Cham, 2016 (pdf).

Avec Y. Benoist, On the regularity of stationary measures, Israel Journal of Mathematics 226 (2018), 1-14 (pdf).

Additive representations of tree lattices 1. The smooth case (pdf).

Avec I. Grama et H. Xiao, A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group (pdf).

Avec Y. Benoist, How far are p-adic Lie groups from algebraic groups?, (pdf).


Doctoral Program Math Requirements

This page describes the Department’s expectations about students’ mathematical knowledge at two points in their course of study: upon entering the program, and before taking the preliminary exams at the end of the first year.

I. Incoming Graduate Students

The department requires incoming students to have completed the following courses before starting graduate study:

  • A three-course sequence in calculus, including multivariate calculus
  • One course in linear algebra
  • One course in mathematical statistics (doctoral students)

While most of the material in these classes is useful for graduate work in economics, it is not always clear to incoming students which topics covered in these classes are most important to review before graduate coursework begins. We list five key topics below. Two useful references on these topics are:

[DS] M. H. Degroot and M. J. Schervish, Probability and Statistics. 3rd ed. Boston: Addison-Wesley, 2002.

[SB] C. P. Simon and L. E. Blume, Mathematics for Economists. New York: Norton, 1994.

The first item summarizes the contents of a standard mathematical statistics class:

1. Basic probability theory (discrete and continuous random variables, conditional probability, expectations, the weak law of large numbers, the Central Limit Theorem) and basic mathematical statistics (point estimation, interval estimation, hypothesis testing). [DS, various chapters].

The next three should be familiar from the calculus course sequence:

2. Geometric representation of vectors and functions in n-dimensional space [SB, Ch. 10 and Sec. 13.2].

3. Differentiation of multivariate functions [SB, Sec. 14.1-14.6].

4. Unconstrained optimization, and equality constrained optimization via the Lagrange multiplier method [SB, Ch. 17 and Sec. 18.2].

The last item is not covered in all undergraduate calculus sequences. Still, we advise students to study this topic before classes begin.

5. Basic convexity [SB, Sec. 21.1-21.3].

Are you a UW-Madison student looking for a recommended set of courses? Take a look at the academic path document provided by the Department of Economics Undergraduate Program. The set of courses suggested to prepare for graduate school in economics will help you acquire the mathematical knowledge that can ease the transition into a Graduate Program in Economics. You can see the description of the courses at the Department of Mathematics’ Undergraduate Course Description page.

II. Students completing first-year doctoral coursework

By the time first-year course work is over, students have learned a wide range of mathematical techniques for economics analysis. Below, we list some fundamental topics that students should master before taking the micro and macro prelims. As additional references, we mention two standard textbooks for the macro curriculum and one for the micro curriculum:

[S] T. J. Sargent, Macroeconomic Theory. 2nd ed. San Diego: Academic Press, 1987.

[SL] N. L. Stokey and R. E. Lucas, Recursive Methods in Economic Dynamics. Cambridge: Harvard University Press, 1995.

N.B.: The Department does not merely expect students to be able to recite the basic results from each of the areas listed below. Rather, we expect students to possess a working knowledge of each topic on the list. This means, for example, that students should be able to recognize situations in which the various techniques are relevant, and should understand how to put the techniques to use.

Two broad, fundamental topics from the first year theory sequences are:

1. Constrained optimization, the Kuhn-Tucker conditions, and concave maximization [SB, Ch. 18, 19, 21, amd 22 MWG, Sec. M.K.].

2. Comparative statics, implicit differentiation, and the implicit function theorem [SB, Ch. 15 and 22 MWG, Sec. M.E. and M.K.].

Two topics that are especially important for the macro sequence are:

3. Mathematics for time series analysis: difference equations, linear regression, Wold representation for time series, spectral analysis [S, Ch. 9-11].

4. Dynamic programming [SL, various chapters].

Two narrowly defined topics closely related to items 1 and 2 are:

5. Homogeneous functions [SB, Sec 20.1 MWG, Sec. M.B].

6. The Envelope Theorem [SB, Sec. 19.2 MWG, Sec. M.L].

Finally, four important “theoretical” topics are:


Formal Statement of the Theorem

The central limit theorem most often applies to a situation in which the variables being averaged have identical probability distribution functions, so the distribution in question is an average measurement over a large number of trials--for example, flipping a coin, rolling a die, or observing the output of a random number generator. There are generalizations of the theorem to other situations, but this wiki will concentrate on the standard applications.

First, the formal statement requires a definition of "converging in distribution" which formalizes the qualitative behavior that the averages get closer and closer to the normal distribution as n n n increases:


3.1 Law of Large Numbers (Chapter 13)

The Law of Large Numbers (LLN) states that if you take independent samples from a well-behaved probability distribution, their sample mean converges in probability to the true mean. Specifically, if (X_<1>,ldots,X_oversetF) and (EX_ <1>= mu) and ( ext(X_<1>) = sigma^ <2>< infty) and (ar_ = (1/n)sum_^X_) then [ ar_overset

< o>mu ] as (n oinfty) . We will investigate the LLN computationally through simulations.

Load the packages we need:

Consider (X_<1>,ldots,X_overset ext(alpha,eta)) . Their mean is (EX_ <1>= alpha/eta) . If we take a sample from this distribution and calculate (ar_) , we should get something close to (alpha/eta) . If we make (n) larger, we should get something closer. The LLN tells us that we can always make (n) large enough such that we get as close as we want with as high a probability as we want.

To simulate from a ( ext(alpha,eta)) distribution in , use the rgamma function:

We can plot the density of the gamma sample mean for various (n) . These densities should have mean (alpha/eta = 2/1 = 2) .

Exercise: if (X_<1>,ldots,X_overset ext(alpha_,eta)) then for any (b) , (b imes sum_^X_ sim extleft(sum_^alpha_,eta/b ight)) . Use this to derive the distribution of (ar) and (hence) explain what the below code is doing.

This is the left side of Figure 13.1 in MIPS:

The density of the sample mean appears centred at (2) and also appears to concentrate around (2) . The LLN says that the sample mean should converge in probability to (2) in this example which would correspond to a density which has a point mass at (2) and is zero everywhere else. It appears like this might be happening.

We can get a better look at “where the sample mean is going” by doing the following:

Simulate a ( ext(2,1)) sample of size (n) for some really big (n) ,

Calculate the the running average of this: for each (m = 1,ldots,n) , compute (ar_ = (1/m)sum_^X_) ,

Plot (ar_) against (m) . (ar_) should get closer and closer to (2) as (m o n) (…probably).

Is this what you expected to see?

Exercise: What happens when you increase the number? Try it for (n = 1,000, n = 10,000) , and so on.

What happens if you break the assumptions of the LLN, that ( ext(X_<1>) < infty) ? The Cauchy distribution has some of the worst mathematical properties of any “named” probability distribution, and so is often used to illustrate what happens when assumptions of theorems aren’t met. It has (EX^ = infty) for any (rgeq 1) so has no finite mean, variance, etc.

Let’s recreate the running average experiment for a random sample (X_<1>,ldots,X_overset ext(mu,sigma)) :

Is this what you expected to see? Should you expect to see anything in particular?

Exercise: repeat this for the Pareto distribution. You can simulate from the Pareto distribution using the rpareto function in the actuar package. Type install.packages("actuar") and then actuar::rpareto . Type ?actuar::rpareto to get help on using this function. Figuring out how to use the function is part of the exercise. Read up on the pareto distribution in MIPS or on Wikipedia or whatever. What do you expect this plot to show? Reading up on the distribution is also part of the exercise.

3.1.1 Extended example: the probability of heads

As an extended example, consider trying to figure out what the probability of heads is for a fair coin, just based on flipping the coin a bunch of times. We can use the LLN to address this challenge.

Let (X) be a random variable which takes values (0) and (1) if the coin comes up tails or heads on any given flip. The specific event that we are interested in is whether the coin comes up heads on any given flip. (p = ext

(X = 1)) is hence the probability of heads and (1-p = ext

(X = 0)) is the probability of tails. We also have (EX = 0 imes (1-p) + 1 imes p = p) . We can hence see what (p) must be by flipping the coin a bunch of times, computing the sample proportion of flips that are heads, and then seeing where this sample proportion appears to be “going.”

Exercise: create a plot of the running average of sample proportions of heads, similar to the above plots for the Gamma and Cauchy. How many times do you think you need to flip the coin before the result is an accurate estimate? Does this change for different values of (p) ?


Lecture Notes

Note: A more recent version of this course, taught by Prof. Dmitry Panchenko at Texas A&M University, is available here. Updated Lecture Notes include some new material and many more exercises.

Stopping times, Wald's identity

Markov property, another proof of SLLN

Lindeberg's central limit theorem

Levy's equivalence theorem, three series theorem

Levy's equivalence theorem, three series theorem (cont.)

Martingales, Doob's decomposition

Convergence on metric spaces, Portmanteau theorem

Laws of Brownian motion at stopping times

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Contents

Classical CLT Edit

Lyapunov CLT Edit

In practice it is usually easiest to check Lyapunov's condition for δ = 1 < extstyle delta =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 Edit

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 ε > 0

converges towards the standard normal distribution N ( 0 , 1 ) < extstyle >(0,1)> .

Multidimensional CLT Edit

The multivariate central limit theorem states that

The rate of convergence is given by the following Berry–Esseen type result:

It is unknown whether the factor d 1 / 4 < extstyle d^<1/4>> is necessary. [9]

Generalized theorem Edit

The central limit theorem states that the sum of a number of independent and identically distributed random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with a power-law tail (Paretian tail) distributions decreasing as | x | − α − 1 < extstyle <|x|>^<-alpha -1>> where 0 < α < 2 < extstyle 0<alpha <2>(and therefore having infinite variance) will tend to a stable distribution f ( x α , 0 , c , 0 ) < extstyle f(xalpha ,0,c,0)>as the number of summands grows. [10] [11] If α > 2 < extstyle alpha >2>then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution. [12]

CLT under weak dependence Edit

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 < extstyle alpha (n) o 0>where α ( n ) < extstyle alpha (n)>is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is: [13]

where the series converges absolutely.

Martingale difference CLT Edit

Proof of classical CLT Edit

The central limit theorem has a proof using characteristic functions. [17] It is similar to the proof of the (weak) law of large numbers.

Convergence to the limit Edit

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. [ citation needed ]

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment E ⁡ [ ( X 1 − μ ) 3 ] < extstyle operatorname left[(X_<1>-mu )^<3> ight]> exists and is finite, then the speed of convergence is at least on the order of 1 / n < extstyle 1/>> (see Berry–Esseen theorem). Stein's method [18] can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics. [19]

The convergence to the normal distribution is monotonic, in the sense that the entropy of Z n < extstyle Z_> increases monotonically to that of the normal distribution. [20]

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 realizations 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 Edit

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 f ( n ) < extstyle f(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 f(n) changes in its leading term.

Informally, one can say: " f(n) grows approximately as a1φ1(n) ". Taking the difference between f(n) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about f(n) :

Here one can say that the difference between the function and its approximation grows approximately as a2φ2(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 happens when the sum, Sn , of independent identically distributed random variables, X1, …, Xn , is studied in classical probability theory. [ citation needed ] If each Xi has finite mean μ , then by the law of large numbers, Sn / nμ . [21] If in addition each Xi has finite variance σ 2 , then by the central limit theorem,

where ξ is distributed as N(0,σ 2 ) . This provides values of the first two constants in the informal expansion

In the case where the Xi do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:

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 √ n log log n , intermediate in size between n of the law of large numbers and √ n of the central limit theorem, provides a non-trivial limiting behavior.

Alternative statements of the theorem Edit

Density functions Edit

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 [24] for a particular local limit theorem for sums of independent and identically distributed random variables.

Characteristic functions Edit

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 Edit

Let Sn be the sum of n random variables. Many central limit theorems provide conditions such that Sn / √ Var( Sn ) converges in distribution to N(0,1) (the normal distribution with mean 0, variance 1) as n → ∞ . In some cases, it is possible to find a constant σ 2 and function f(n) such that Sn /(σ √ n⋅f ( n ) ) converges in distribution to N(0,1) as n → ∞ .

Products of positive random variables Edit

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. [26]

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 Edit

Theorem. There exists a sequence εn ↓ 0 for which the following holds. Let n ≥ 1 , and let random variables X1, …, Xn have a log-concave joint density f such that f(x1, …, xn) = f(| x1 |, …, | xn |) for all x1, …, xn , and E(X 2
k ) = 1 for all k = 1, …, n . Then the distribution of

X 1 + ⋯ + X n n +cdots +X_>>>>

is εn -close to N(0,1) in the total variation distance. [27]

These two εn -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(x1, …, xn) = const · exp(−(| x1 | α + ⋯ + | xn | α ) β ) where α > 1 and αβ > 1 . If β = 1 then f(x1, …, xn) factorizes into const · exp (−| x1 | α ) … exp(−| xn | α ), which means X1, …, Xn are independent. In general, however, they are dependent.

The condition f(x1, …, xn) = f(| x1 |, …, | xn |) ensures that X1, …, Xn are of zero mean and uncorrelated [ citation needed ] still, they need not be independent, nor even pairwise independent. [ citation needed ] By the way, pairwise independence cannot replace independence in the classical central limit theorem. [28]

Lacunary trigonometric series Edit

  • nk satisfy the lacunarity condition: there exists q > 1 such that nk + 1qnk for all k ,
  • rk are such that r 1 2 + r 2 2 + ⋯ = ∞ and r k 2 r 1 2 + ⋯ + r k 2 → 0 , ^<2>+r_<2>^<2>+cdots =infty quad < ext< and >>quad ^<2>>^<2>+cdots +r_^<2>>> o 0,>
  • 0 ≤ ak < 2π .

Gaussian polytopes Edit

Theorem: Let A1, …, An be independent random points on the plane 2 each having the two-dimensional standard normal distribution. Let Kn be the convex hull of these points, and Xn the area of Kn Then [33]

X n − E ( X n ) Var ⁡ ( X n ) -mathbb (X_)> (X_)>>>>

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

The same also holds in all dimensions greater than 2.

The polytope Kn is called a 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. [34]

Linear functions of orthogonal matrices Edit

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,) see Rotation matrix#Uniform random rotation matrices.

Subsequences Edit

Random walk on a crystal lattice Edit

The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures. [37] [38]

Simple example Edit

50) and the mean sample standard deviation divided by the square root of the sample size (

28.87/ √ n ), which is called the standard deviation of the mean (since it refers to the spread of sample means).

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.

Real applications Edit

Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem. [39] One source [40] states the following examples:

  • The probability distribution for total distance covered in a random walk (biased or unbiased) will tend toward a normal distribution.
  • Flipping many 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 many 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.

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 Edit

Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem. [41]

Dutch mathematician Henk Tijms writes: [42]

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 described the Central Limit Theorem in this way: [43]

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 marshalled 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. [44] [45] 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". [45] The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya [44] in 1920 translates as follows.

The occurrence of the Gaussian probability density 1 = ex 2 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. [46] 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. [47] Le Cam describes a period around 1935. [45] Bernstein [48] presents 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 not published. [49]


A central limit theorem is a theorem that states the standard deviation of a sample is equal to the standard deviation of the population divided by the square root of the sample size.

How to calculate standard deviation using the central limit theorem

    First, determine the standard deviation of the population

Using the formula for standard deviation, calculate this value for the entire population.

This is the total size of the sample, denoted as n in the calculator above.

Determine the standard deviation of the sample using the formula above and the values from steps 1 and 2.

The central limit theorem states that the population and sample mean of a data set are so close that they can be considered equal. That is the X = u. This simplifies the equation for calculating the sample standard deviation to the equation mentioned above.