In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the distribution of the average will be closely approximated by a normal distribution. A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series. (In the limit of an infinite number of flips, it will equal a normal curve.)
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for nonidentical distributions or for nonindependent observations, given that they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is now known as the de Moivre–Laplace theorem.
In more general usage, a central limit theorem is any of a set of weakconvergence theorems in probability theory. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as x^{−α − 1} where 0 < α < 2 (and therefore having infinite variance) will tend to an alphastable distribution with stability parameter (or index of stability) of α as the number of variables grows.^{[1]}
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✪ The Central Limit Theorem
Transcription
In this video, I want to talk about what is easily one of the most fundamental and profound concepts in statistics and maybe in all of mathematics. And that's the central limit theorem. And what it tells us is we can start off with any distribution that has a welldefined mean and variance and if it has a welldefined variance, it has a welldefined standard deviation. And it could be a continuous distribution or a discrete one. I'll draw a discrete one, just because it's easier to imagine, at least for the purposes of this video. So let's say I have a discrete probability distribution function. And I want to be very careful not to make it look anything close to a normal distribution. Because I want to show you the power of the central limit theorem. So let's say I have a distribution. Let's say it could take on values 1 through 6. 1, 2, 3, 4, 5, 6. It's some kind of crazy dice. It's very likely to get a one. Let's say it's impossible well, let me make that a straight line. You have a very high likelihood of getting a 1. Let's say it's impossible to get a 2. Let's say it's an OK likelihood of getting a 3 or a 4. Let's say it's impossible to get a 5. And let's say it's very likely to get a 6 like that. So that's my probability distribution function. If I were to draw a mean this the symmetric, so maybe the mean would be something like that. The mean would be halfway. So that would be my mean right there. The standard deviation maybe would look it would be that far and that far above and below the mean. But that's my discrete probability distribution function. Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it. But I'm going to average the samples and then look at those samples and see the frequency of the averages that I get. And when I say average, I mean the mean. Let me define something. Let's say my sample size and I could put any number here. But let's say first off we try a sample size of n is equal to 4. And what that means is I'm going to take four samples from this. So let's say the first time I take four samples so my sample sizes is four let's say I get a 1. Let's say I get another 1. And let's say I get a 3. And I get a 6. So that right there is my first sample of sample size 4. I know the terminology can get confusing. Because this is the sample that's made up of four samples. But then when we talk about the sample mean and the sampling distribution of the sample mean, which we're going to talk more and more about over the next few videos, normally the sample refers to the set of samples from your distribution. And the sample size tells you how many you actually took from your distribution. But the terminology can be very confusing, because you could easily view one of these as a sample. But we're taking four samples from here. We have a sample size of four. And what I'm going to do is I'm going to average them. So let's say the mean I want to be very careful when I say average. The mean of this first sample of size 4 is what? 1 plus 1 is 2. 2 plus 3 is 5. 5 plus 6 is 11. 11 divided by 4 is 2.75. That is my first sample mean for my first sample of size 4. Let me do another one. My second sample of size 4, let's say that I get a 3, a 4. Let's say I get another 3. And let's say I get a 1. I just didn't happen to get a 6 that time. And notice I can't get a 2 or a 5. It's impossible for this distribution. The chance of getting a 2 or 5 is 0. So I can't have any 2s or 5s over here. So for the second sample of sample size 4, my second sample mean is going to be 3 plus 4 is 7. 7 plus 3 is 10 plus 1 is 11. 11 divided by 4, once again, is 2.75. Let me do one more, because I really want to make it clear what we're doing here. So I do one more. Actually, we're going to do a gazillion more. But let me just do one more in detail. So let's say my third sample of sample size 4 so I'm going to literally take 4 samples. So my sample is made up of 4 samples from this original crazy distribution. Let's say I get a 1, a 1, and a 6 and a 6. And so my third sample mean is going to be 1 plus 1 is 2. 2 plus 6 is 8. 8 plus 6 is 14. 14 divided by 4 is 3 and 1/2. And as I find each of these sample means so for each of my samples of sample size 4, I figure out a mean. And as I do each of them, I'm going to plot it on a frequency distribution. And this is all going to amaze you in a few seconds. So I plot this all on a frequency distribution. So I say, OK, on my first sample, my first sample mean was 2.75. So I'm plotting the actual frequency of the sample means I get for each sample. So 2.75, I got it one time. So I'll put a little plot there. So that's from that one right there. And the next time, I also got a 2.75. That's a 2.75 there. So I got it twice. So I'll plot the frequency right there. Then I got a 3 and 1/2. So all the possible values, I could have a three, I could have a 3.25, I could have a 3 and 1/2. So then I have the 3 and 1/2, so I'll plot it right there. And what I'm going to do is I'm going to keep taking these samples. Maybe I'll take 10,000 of them. So I'm going to keep taking these samples. So I go all the way to S 10,000. I just do a bunch of these. And what it's going to look like over time is each of these I'm going to make it a dot, because I'm going to have to zoom out. So if I look at it like this, over time it still has all the values that it might be able to take on, 2.75 might be here. So this first dot is going to be this one right here is going to be right there. And that second one is going to be right there. Then that one at 3.5 is going to look right there. But I'm going to do it 10,000 times. Because I'm going to have 10,000 dots. And let's say as I do it, I'm going just keep plotting them. I'm just going to keep plotting the frequencies. I'm just going to keep plotting them over and over and over again. And what you're going to see is, as I take many, many samples of size 4, I'm going to have something that's going to start kind of approximating a normal distribution. So each of these dots represent an incidence of a sample mean. So as I keep adding on this column right here, that means I kept getting the sample mean 2.75. So over time. I'm going to have something that's starting to approximate a normal distribution. And that is a neat thing about the central limit theorem. So an orange, that's the case for n is equal to 4. This was a sample size of 4. Now, if I did the same thing with a sample size of maybe 20 so in this case, instead of just taking 4 samples from my original crazy distribution, every sample I take 20 instances of my random variable, and I average those 20. And then I plot the sample mean on here. So in that case, I'm going to have a distribution that looks like this. And we'll discuss this in more videos. But it turns out if I were to plot 10,000 of the sample means here, I'm going to have something that, two things it's going to even more closely approximate a normal distribution. And we're going to see in future videos, it's actually going to have a smaller well, let me be clear. It's going to have the same mean. So that's the mean. This is going to have the same mean. So it's going to have a smaller standard deviation. Well, I should plot these from the bottom because you kind of stack it. One you get one, then another instance and another instance. But this is going to more and more approach a normal distribution. So this is what's super cool about the central limit theorem. As your sample size becomes larger or you could even say as it approaches infinity. But you really don't have to get that close to infinity to really get close to a normal distribution. Even if you have a sample size of 10 or 20, you're already getting very close to a normal distribution, in fact about as good an approximation as we see in our everyday life. But what's cool is we can start with some crazy distribution. This has nothing to do with a normal distribution. This was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these, instead of four here, and average them and then plot that average, the frequency of it, then we take 100 again, average them, take the mean, plot that again, and if we do that a bunch of times, in fact, if we were to do that an infinite time, we would find that we, especially if we had an infinite sample size, we would find a perfect normal distribution. That's the crazy thing. And it doesn't apply just to taking the sample mean. Here we took the sample mean every time. But you could have also taken the sample sum. The central limit theorem would have still applied. But that's what's so super useful about it. Because in life, there's all sorts of processes out there, proteins bumping into each other, people doing crazy things, humans interacting in weird ways. And you don't know the probability distribution functions for any of those things. But what the central limit theorem tells us is if we add a bunch of those actions together, assuming that they all have the same distribution, or if we were to take the mean of all of those actions together, and if we were to plot the frequency of those means, we do get a normal distribution. And that's frankly why the normal distribution shows up so much in statistics and why, frankly, it's a very good approximation for the sum or the means of a lot of processes. Normal distribution. What I'm going to show you in the next video is I'm actually going to show you that this is a reality, that as you increase your sample size, as you increase your n, and as you take a lot of sample means, you're going to have a frequency plot that looks very, very close to a normal distribution.
Contents
Independent sequences
Classical CLT
Let {X_{1}, …, X_{n}} be a random sample of size n—that is, a sequence of independent and identically distributed (i.i.d.) random variables drawn from a distribution of expected value given by µ and finite variance given by σ^{2}. Suppose we are interested in the sample average
of these random variables. By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n → ∞. The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More precisely, it states that as n gets larger, the distribution of the difference between the sample average S_{n} and its limit µ, when multiplied by the factor √n (that is √n(S_{n} − µ)), approximates the normal distribution with mean 0 and variance σ^{2}. For large enough n, the distribution of S_{n} is close to the normal distribution with mean µ and variance σ^{2}/n. The usefulness of the theorem is that the distribution of √n(S_{n} − µ) approaches normality regardless of the shape of the distribution of the individual X_{i}. Formally, the theorem can be stated as follows:
Lindeberg–Lévy CLT. Suppose {X_{1}, X_{2}, …} is a sequence of i.i.d. random variables with E[X_{i}] = µ and Var[X_{i}] = σ^{2} < ∞. Then as n approaches infinity, the random variables √n(S_{n} − µ) converge in distribution to a normal N(0,σ^{2}):^{[3]}
In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n(S_{n} − µ) converge pointwise to the cdf of the N(0, σ^{2}) distribution: for every real number z,
where Φ(z) is the standard normal cdf evaluated at z. Note that the convergence is uniform in z in the sense that
where sup denotes the least upper bound (or supremum) of the set.^{[4]}
Lyapunov CLT
The theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the central limit theorem the random variables X_{i} have to be independent, but not necessarily identically distributed. The theorem also requires that random variables X_{i} have moments of some order (2 + δ), and that the rate of growth of these moments is limited by the Lyapunov condition given below.
Lyapunov CLT.^{[5]} Suppose {X_{1}, X_{2}, …} is a sequence of independent random variables, each with finite expected value μ_{i} and variance σ^{2}
_{i}. Define
If for some δ > 0, Lyapunov’s condition
is satisfied, then a sum of X_{i} − μ_{i}/s_{n} converges in distribution to a standard normal random variable, as n goes to infinity:
In practice it is usually easiest to check Lyapunov's condition for δ = 1.
If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
Lindeberg CLT
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920).
Suppose that for every ε > 0
where 1_{{…}} is the indicator function. Then the distribution of the standardized sums
converges towards the standard normal distribution N(0,1).
Multidimensional CLT
Proofs that use characteristic functions can be extended to cases where each individual X_{i} is a random vector in ℝ^{k}, with mean vector μ = E(X_{i}) and covariance matrix Σ (among the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done componentwise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.^{[6]}
Let
be the kvector. The bold in X_{i} means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be
and the average is
and therefore
The multivariate central limit theorem states that
where the covariance matrix Σ is equal to
The rate of convergence is given by the following Berry–Esseen type result:
Theorem.^{[7]} Let be independent valued random vectors, each having mean zero. Write and assume is invertible. Let be a dimensional Gaussian with the same mean and covariance matrix as . Then for all convex sets ,
where is a universal constant, , and denotes the Euclidean norm on .
It is unknown whether the factor is necessary.^{[8]}
Generalized theorem
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 powerlaw tail (Paretian tail) distributions decreasing as x^{−α − 1} where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution f(x;α,0,c,0) as the number of summands grows.^{[9]}^{[10]} If α > 2 then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution.^{[11]}
Dependent processes
CLT under weak dependence
A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called αmixing) defined by α(n) → 0 where α(n) is socalled strong mixing coefficient.
A simplified formulation of the central limit theorem under strong mixing is:^{[12]}
Theorem. Suppose that X_{1}, X_{2}, … is stationary and αmixing with α_{n} = O(n^{−5}) and that E(X_{n}) = 0 and E(X^{12}
_{n}) < ∞. Denote S_{n} = X_{1} + … + X_{n}, then the limit
exists, and if σ ≠ 0 then S_{n}/σ√n converges in distribution to N(0,1).
In fact,
where the series converges absolutely.
The assumption σ ≠ 0 cannot be omitted, since the asymptotic normality fails for X_{n} = Y_{n} − Y_{n − 1} where Y_{n} are another stationary sequence.
There is a stronger version of the theorem:^{[13]} the assumption E(X^{12}
_{n}) < ∞ is replaced with E(X_{n}^{2 + δ}) < ∞, and the assumption α_{n} = O(n^{−5}) is replaced with
Existence of such δ > 0 ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).
Martingale difference CLT
Theorem. Let a martingale M_{n} satisfy
 in probability as n → ∞,
 for every ε > 0, as n → ∞,
then M_{n}/√n converges in distribution to N(0,1) as n → ∞.^{[14]}^{[15]}
Caution: The restricted expectation^{[clarification needed]} E(X ; A) should not be confused with the conditional expectation E(X  A) = E(X ; A)/P(A).
Remarks
Proof of classical CLT
The central limit theorem has a simple proof using characteristic functions.^{[16]} It is similar to the proof of the (weak) law of large numbers.
Assume {X_{1}, …, X_{n}} are independent and identically distributed random variables, each with mean µ and finite variance σ^{2}. The sum X_{1} + … + X_{n} has mean nµ and variance nσ^{2}. Consider the random variable
where in the last step we defined the new random variables Y_{i} = X_{i} − μ/σ, each with zero mean and unit variance (var(Y) = 1). The characteristic function of Z_{n} is given by
where in the last step we used the fact that all of the Y_{i} are identically distributed. The characteristic function of Y_{1} is, by Taylor's theorem,
where o(t^{2}) is "little o notation" for some function of t that goes to zero more rapidly than t^{2}. By the limit of the exponential function (e^{x}= lim(1 + x/n)^{n}), the characteristic function of Z_{n} equals
Note that all of the higher order terms vanish in the limit n → ∞. The right hand side equals the characteristic function of a standard normal distribution N(0,1), which implies through Lévy's continuity theorem that the distribution of Z_{n} will approach N(0,1) as n → ∞. Therefore, the sum X_{1} + … + X_{n} will approach that of the normal distribution N(nµ,nσ^{2}), and the sample average
converges to the normal distribution N(µ,σ^{2}/n), from which the central limit theorem follows.
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.^{[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}) exists and is finite, then the speed of convergence is at least on the order of 1/√n (see Berry–Esseen theorem). Stein's method^{[17]} can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.^{[18]}
The convergence to the normal distribution is monotonic, in the sense that the entropy of Z_{n} increases monotonically to that of the normal distribution.^{[19]}
The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realisations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity, this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
Relation to the law of large numbers
The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behaviour 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):
Dividing both parts by φ_{1}(n) and taking the limit will produce a_{1}, the coefficient of the highestorder 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 a_{1}φ_{1}(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 a_{2}φ_{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 happens when the sum, S_{n}, of independent identically distributed random variables, X_{1}, …, X_{n}, is studied in classical probability theory.^{[citation needed]} If each X_{i} has finite mean μ, then by the law of large numbers, S_{n}/n → μ.^{[20]} If in addition each X_{i} 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 X_{i} do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
or informally
Distributions Ξ which can arise in this way are called stable.^{[21]} Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor b_{n} may be proportional to n^{c}, for any c ≥ 1/2; it may also be multiplied by a slowly varying function of n.^{[11]}^{[22]}
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 nontrivial 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. 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^{[23]} for a particular local limit theorem for sums of independent and identically distributed random variables.
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. 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
Let S_{n} be the sum of n random variables. Many central limit theorems provide conditions such that S_{n}/√Var(S_{n}) 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 S_{n}/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.
Lemma.^{[24]} Suppose is a sequence of realvalued and strictly stationary random variables with for all , , and . Construct
 If is absolutely convergent, , and then as where .
 If in addition and converges in distribution to as then also converges in distribution to as .
Extensions
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 lognormal 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 lognormal 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 squareintegrable.^{[25]}
Beyond the classical framework
Asymptotic normality, that is, convergence to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
Convex body
Theorem. There exists a sequence ε_{n} ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X_{1}, …, X_{n} have a logconcave joint density f such that f(x_{1}, …, x_{n}) = f(x_{1}, …, x_{n}) for all x_{1}, …, x_{n}, and E(X^{2}
_{k}) = 1 for all k = 1, …, n. Then the distribution of
is ε_{n}close to N(0,1) in the total variation distance.^{[26]}
These two ε_{n}close distributions have densities (in fact, logconcave 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 logconcave 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}, …, x_{n}) = const · exp( − (x_{1}^{α} + … + x_{n}^{α})^{β}) where α > 1 and αβ > 1. If β = 1 then f(x_{1}, …, x_{n}) factorizes into const · exp (−x_{1}^{α}) … exp(−x_{n}^{α}), which means X_{1}, …, X_{n} are independent. In general, however, they are dependent.
The condition f(x_{1}, …, x_{n}) = f(x_{1}, …, x_{n}) ensures that X_{1}, …, X_{n} 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.^{[27]}
Here is a Berry–Esseen type result.
Theorem. Let X_{1}, …, X_{n} satisfy the assumptions of the previous theorem, then ^{[28]}
for all a < b; here C is a universal (absolute) constant. Moreover, for every c_{1}, …, c_{n} ∈ ℝ such that c^{2}
_{1} + … + c^{2}
_{n} = 1,
The distribution of X_{1} + … + X_{n}/√n need not be approximately normal (in fact, it can be uniform).^{[29]} However, the distribution of c_{1}X_{1} + … + c_{n}X_{n} is close to N(0,1) (in the total variation distance) for most vectors (c_{1}, …, c_{n}) according to the uniform distribution on the sphere c^{2}
_{1} + … + c^{2}
_{n} = 1.
Lacunary trigonometric series
Theorem (Salem–Zygmund): Let U be a random variable distributed uniformly on (0,2π), and X_{k} = r_{k} cos(n_{k}U + a_{k}), where
 n_{k} satisfy the lacunarity condition: there exists q > 1 such that n_{k + 1} ≥ qn_{k} for all k,
 r_{k} are such that
 0 ≤ a_{k} < 2π.
Then^{[30]}^{[31]}
converges in distribution to N(0, 1/2).
Gaussian polytopes
Theorem: Let A_{1}, …, A_{n} be independent random points on the plane ℝ^{2} each having the twodimensional standard normal distribution. Let K_{n} be the convex hull of these points, and X_{n} the area of K_{n} Then^{[32]}
converges in distribution to N(0,1) as n tends to infinity.
The same also holds in all dimensions greater than 2.
The polytope K_{n} 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.^{[33]}
Linear functions of orthogonal matrices
A linear function of a matrix M is a linear combination of its elements (with given coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product.
A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group O(n,ℝ); see Rotation matrix#Uniform random rotation matrices.
Theorem. Let M be a random orthogonal n × n matrix distributed uniformly, and A a fixed n × n matrix such that tr(AA*) = n, and let X = tr(AM). Then^{[34]} the distribution of X is close to N(0,1) in the total variation metric up to^{[clarification needed]} 2√3/n − 1.
Subsequences
Theorem. Let random variables X_{1}, X_{2}, … ∈ L_{2}(Ω) be such that X_{n} → 0 weakly in L_{2}(Ω) and X^{}
_{n} → 1 weakly in L_{1}(Ω). Then there exist integers n_{1} < n_{2} < … such that
converges in distribution to N(0,1) as k tends to infinity.^{[35]}
Random walk on a crystal lattice
The central limit theorem may be established for the simple random walk on a crystal lattice (an infinitefold abelian covering graph over a finite graph), and is used for design of crystal structures. ^{[36]}^{[37]}
Applications and examples
Simple example
A simple example of the central limit theorem is rolling a large number of identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since realworld 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 largesample statistics to the normal distribution in controlled experiments.
Real applications
Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem.^{[38]} One source^{[39]} 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 a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).
From another viewpoint, the central limit theorem explains the common appearance of the "bell curve" in density estimates applied to real world data. In cases like electronic noise, examination grades, and so on, we can often regard a single measured value as the weighted average of a large number of small effects. Using generalisations of the central limit theorem, we can then see that this would often (though not always) produce a final distribution that is approximately normal.
In general, the more a measurement is like the sum of independent variables with equal influence on the result, the more normality it exhibits. This justifies the common use of this distribution to stand in for the effects of unobserved variables in models like the linear model.
Regression
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 a large number of 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
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.^{[40]}
History
Dutch mathematician Henk Tijms writes:^{[41]}
The central limit theorem has an interesting history. The first version of this theorem was postulated by the Frenchborn 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 PierreSimon 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:^{[42]}
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 selfeffacement, 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.^{[43]}^{[44]} 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".^{[44]} The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya^{[43]} in 1920 translates as follows.
The occurrence of the Gaussian probability density 1 = e^{−x2} in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is wellknown, 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.^{[45]} 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.^{[46]} Le Cam describes a period around 1935.^{[44]} Bernstein^{[47]} 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.
Through the 1930s, progressively more general proofs of the Central Limit Theorem were presented. Many natural systems were found to exhibit Gaussian distributions—a typical example being height distributions for humans. When statistical methods such as analysis of variance became established in the early 1900s, it became increasingly common to assume underlying Gaussian distributions.^{[48]}
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]}^{[50]}^{[51]}
See also
 Asymptotic equipartition property
 Asymptotic distribution
 Bates distribution
 Benford's law – Result of extension of CLT to product of random variables.
 Berry–Esseen theorem
 Central limit theorem for directional statistics – Central limit theorem applied to the case of directional statistics
 Delta method – to compute the limit distribution of a function of a random variable.
 Erdős–Kac theorem – connects the number of prime factors of an integer with the normal probability distribution
 Fisher–Tippett–Gnedenko theorem – limit theorem for extremum values (such as max{X_{n}})
 Irwin–Hall distribution
 Markov chain central limit theorem
 Normal distribution
 Tweedie convergence theorem – A theorem that can be considered to bridge between the central limit theorem and the Poisson convergence theorem^{[52]}
 Normal distribution
Notes
 ^ Voit, Johannes (2003). The Statistical Mechanics of Financial Markets. SpringerVerlag. p. 124. ISBN 3540009787.
 ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 10.
 ^ Billingsley (1995, p. 357)
 ^ Bauer (2001, Theorem 30.13, p.199)
 ^ Billingsley (1995, p.362)
 ^ Van der Vaart, A. W. (1998). Asymptotic statistics. New York: Cambridge University Press. ISBN 9780521496032. LCCN 98015176.
 ^ Ryan O’Donnell (2014, Theorem 5.38) http://www.contrib.andrew.cmu.edu/~ryanod/?p=866
 ^ Bentkus, V. (2005). "A Lyapunovtype Bound in ". Theory Probab. Appl. 49 (2): 311–323. doi:10.1137/S0040585X97981123.
 ^ Voit, Johannes (2003). "Section 5.4.3". The Statistical Mechanics of Financial Markets. Texts and Monographs in Physics. SpringerVerlag. ISBN 3540009787.
 ^ Gnedenko, B. V.; Kolmogorov, A. N. (1954). Limit distributions for sums of independent random variables. Cambridge: AddisonWesley.
 ^ ^{a} ^{b} Uchaikin, Vladimir V.; Zolotarev, V. M. (1999). Chance and stability: stable distributions and their applications. VSP. pp. 61–62. ISBN 9067643017.
 ^ Billingsley (1995, Theorem 27.5)
 ^ Durrett (2004, Sect. 7.7(c), Theorem 7.8)
 ^ Durrett (2004, Sect. 7.7, Theorem 7.4)
 ^ Billingsley (1995, Theorem 35.12)
 ^ "An Introduction to Stochastic Processes in Physics". jhupbooks.press.jhu.edu. Retrieved 20160811.
 ^ Stein, C. (1972). "A bound for the error in the normal approximation to the distribution of a sum of dependent random variables". Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability: 583–602. MR 0402873. Zbl 0278.60026.
 ^ Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). Normal approximation by Stein's method. Springer. ISBN 9783642150067.
 ^ Artstein, S.; Ball, K.; Barthe, F.; Naor, A. (2004), "Solution of Shannon's Problem on the Monotonicity of Entropy", Journal of the American Mathematical Society, 17 (4): 975–982, doi:10.1090/S089403470400459X
 ^ Rosenthal, Jeffrey Seth (2000). A First Look at Rigorous Probability Theory. World Scientific. Theorem 5.3.4, p. 47. ISBN 9810243227.
 ^ Johnson, Oliver Thomas (2004). Information Theory and the Central Limit Theorem. Imperial College Press. p. 88. ISBN 1860944736.
 ^ Borodin, A. N.; Ibragimov, I. A.; Sudakov, V. N. (1995). Limit Theorems for Functionals of Random Walks. AMS Bookstore. Theorem 1.1, p. 8. ISBN 0821804383.
 ^ Petrov, V. V. (1976). Sums of Independent Random Variables. New YorkHeidelberg: SpringerVerlag. ch. 7.
 ^ Hew, Patrick Chisan (2017). "Asymptotic distribution of rewards accumulated by alternating renewal processes". Statistics and Probability Letters. 129: 355–359. doi:10.1016/j.spl.2017.06.027.
 ^ Rempala, G.; Wesolowski, J. (2002). "Asymptotics of products of sums and Ustatistics" (PDF). Electronic Communications in Probability. 7: 47–54. doi:10.1214/ecp.v71046.
 ^ Klartag (2007, Theorem 1.2)
 ^ Durrett (2004, Section 2.4, Example 4.5)
 ^ Klartag (2008, Theorem 1)
 ^ Klartag (2007, Theorem 1.1)
 ^ Zygmund, Antoni (2003) [1959]. Trigonometric Series. Cambridge University Press. vol. II, sect. XVI.5, Theorem 55. ISBN 0521890535.
 ^ Gaposhkin (1966, Theorem 2.1.13)
 ^ Bárány & Vu (2007, Theorem 1.1)
 ^ Bárány & Vu (2007, Theorem 1.2)
 ^ Meckes, Elizabeth (2008). "Linear functions on the classical matrix groups". Transactions of the American Mathematical Society. 360 (10): 5355–5366. arXiv:math/0509441. doi:10.1090/S0002994708044449.
 ^ Gaposhkin (1966, Sect. 1.5)
 ^ Kotani, M.; Sunada, Toshikazu (2003). Spectral geometry of crystal lattices. 338. Contemporary Math. pp. 271–305. ISBN 9780821842690.
 ^ Sunada, Toshikazu (2012). Topological Crystallography – With a View Towards Discrete Geometric Analysis. Surveys and Tutorials in the Applied Mathematical Sciences. 6. Springer. ISBN 9784431541776.
 ^ Dinov, Christou & Sánchez (2008)
 ^ "SOCR EduMaterials Activities GCLT Applications  Socr". Wiki.stat.ucla.edu. 20100524. Retrieved 20170123.
 ^ Marasinghe, M.; Meeker, W.; Cook, D.; Shin, T. S. (Aug 1994). "Using graphics and simulation to teach statistical concepts". Paper presented at the Annual meeting of the American Statistician Association, Toronto, Canada.
 ^ Henk, Tijms (2004). Understanding Probability: Chance Rules in Everyday Life. Cambridge: Cambridge University Press. p. 169. ISBN 0521540364.
 ^ Galton, F. (1889). Natural Inheritance. p. 66.
 ^ ^{a} ^{b} Pólya, George (1920). "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem" [On the central limit theorem of probability calculation and the problem of moments]. Mathematische Zeitschrift (in German). 8 (3–4): 171–181. doi:10.1007/BF01206525.
 ^ ^{a} ^{b} ^{c} Le Cam, Lucien (1986). "The central limit theorem around 1935". Statistical Science. 1 (1): 78–91. doi:10.2307/2245503.
 ^ Hald, Andreas. A History of Mathematical Statistics from 1750 to 1930 (PDF). Gbv.de. chapter 17. ISBN 9780471179122.
 ^ Fischer, Hans (2011). A History of the Central Limit Theorem: From Classical to Modern Probability Theory. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. doi:10.1007/9780387878577. ISBN 9780387878560. MR 2743162. Zbl 1226.60004. (Chapter 2: The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods, Chapter 5.2: The Central Limit Theorem in the Twenties)
 ^ Bernstein, S. N. (1945). "On the work of P. L. Chebyshev in Probability Theory". In Bernstein., S. N. (ed.). Nauchnoe Nasledie P. L. Chebysheva. Vypusk Pervyi: Matematika [The Scientific Legacy of P. L. Chebyshev. Part I: Mathematics] (in Russian). Moscow & Leningrad: Academiya Nauk SSSR. p. 174.
 ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 977. ISBN 1579550088.
 ^ Hodges, Andrew (1983). Alan Turing: The Enigma. London: Burnett Books. pp. 87–88. ISBN 0091521300.
 ^ Zabell, S. L. (2005). Symmetry and Its Discontents: Essays on the History of Inductive Probability. Cambridge University Press. p. 199. ISBN 0521444705.
 ^ Aldrich, John (2009). "England and Continental Probability in the InterWar Years". Electronic Journ@l for History of Probability and Statistics. 5 (2). Section 3.
 ^ Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall. ISBN 9780412997112.
References
 Bárány, Imre; Vu, Van (2007). "Central limit theorems for Gaussian polytopes". Annals of Probability. Institute of Mathematical Statistics. 35 (4): 1593–1621. arXiv:math/0610192. doi:10.1214/009117906000000791.
 Bauer, Heinz (2001). Measure and Integration Theory. Berlin: de Gruyter. ISBN 3110167190.
 Billingsley, Patrick (1995). Probability and Measure (3rd ed.). John Wiley & Sons. ISBN 0471007102.
 Bradley, Richard (2007). Introduction to Strong Mixing Conditions (1st ed.). Heber City, UT: Kendrick Press. ISBN 097404279X.
 Bradley, Richard (2005). "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions" (PDF). Probability Surveys. 2: 107–144. arXiv:math/0511078v1. doi:10.1214/154957805100000104.
 Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008). "Central Limit Theorem: New SOCR Applet and Demonstration Activity". Journal of Statistics Education. ASA. 16 (2). doi:10.1080/10691898.2008.11889560. PMC 3152447.
 Durrett, Richard (2004). Probability: theory and examples (3rd ed.). Cambridge University Press. ISBN 0521765390.
 Gaposhkin, V. F. (1966). "Lacunary series and independent functions". Russian Mathematical Surveys. 21 (6): 1–82. Bibcode:1966RuMaS..21....1G. doi:10.1070/RM1966v021n06ABEH001196..
 Klartag, Bo'az (2007). "A central limit theorem for convex sets". Inventiones Mathematicae. 168: 91–131. arXiv:math/0605014. Bibcode:2007InMat.168...91K. doi:10.1007/s0022200600288.
 Klartag, Bo'az (2008). "A Berry–Esseen type inequality for convex bodies with an unconditional basis". Probability Theory and Related Fields. 145: 1–33. arXiv:0705.0832. doi:10.1007/s0044000801586.
External links
Wikimedia Commons has media related to Central limit theorem. 
 Simplified, stepbystep explanation of the classical Central Limit Theorem. with histograms at every step.
 Handson explanation of the Central Limit Theorem in tutorial videos from Khan Academy, with many examples
 Central Limit Theorem Visualized in D3 interactive HTML5 simulation of flipping coins.
 Hazewinkel, Michiel, ed. (2001) [1994], "Central limit theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Animated examples of the CLT
 Central Limit Theorem interactive simulation to experiment with various parameters
 CLT in NetLogo (Connected Probability — ProbLab) interactive simulation with a variety of modifiable parameters
 General Central Limit Theorem Activity & corresponding SOCR CLT Applet (Select the Sampling Distribution CLT Experiment from the dropdown list of SOCR Experiments)
 Generate sampling distributions in Excel Specify arbitrary population, sample size, and sample statistic.
 MIT OpenCourseWare Lecture 18.440 Probability and Random Variables, Spring 2011, Scott Sheffield Another proof. Retrieved 20120408.
 CAUSEweb.org is a site with many resources for teaching statistics including the Central Limit Theorem
 The Central Limit Theorem by Chris Boucher, Wolfram Demonstrations Project.
 Weisstein, Eric W. "Central Limit Theorem". MathWorld.
 Animations for the Central Limit Theorem by Yihui Xie using the R package animation
 Teaching demonstrations of the CLT: clt.examp function in Greg Snow (2012). TeachingDemos: Demonstrations for teaching and learning. R package version 2.8.