# Central limit theorem

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.

Mathematically, if ${\textstyle X_{1},X_{2},...,X_{n}}$ is a random sample of size ${\textstyle n}$ taken from a population with mean ${\textstyle \mu }$ and finite variance ${\textstyle \sigma ^{2}}$ and if ${\textstyle {\bar {X}}}$ is the sample mean, the limiting form of the distribution of ${\textstyle Z=\left({\frac {{\bar {X}}_{n}-\mu }{\sigma /\surd n}}\right)}$ as ${\textstyle n\to \infty }$, is the standard normal distribution.[1]

For example, suppose that a sample is obtained containing many 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 probability distribution of the average will closely approximate 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 will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution.

The central limit theorem has several 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 non-identical distributions or for non-independent observations, if 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 the de Moivre–Laplace theorem.

## Independent sequences

A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations; see Illustration of the central limit theorem for further details.
Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the Central Limit Theorem.[2]

### Classical CLT

Let ${\textstyle \{X_{1},\ldots ,X_{n}\}}$ be a random sample of size ${\textstyle n}$—that is, a sequence of independent and identically distributed (i.i.d.) random variables drawn from a distribution of expected value given by ${\textstyle \mu }$ and finite variance given by ${\textstyle \sigma ^{2}}$. Suppose we are interested in the sample average

${\displaystyle {\bar {X}}_{n}:={\frac {X_{1}+\cdots +X_{n}}{n}}}$

of these random variables. By the law of large numbers, the sample averages converge in probability and almost surely to the expected value ${\textstyle \mu }$ as ${\textstyle n\to \infty }$. The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number ${\textstyle \mu }$ during this convergence. More precisely, it states that as ${\textstyle n}$ gets larger, the distribution of the difference between the sample average ${\textstyle {\bar {X}}_{n}}$ and its limit ${\textstyle \mu }$, when multiplied by the factor ${\textstyle {\sqrt {n}}}$ (that is ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$), approximates the normal distribution with mean 0 and variance ${\textstyle \sigma ^{2}}$. For large enough n, the distribution of ${\textstyle {\bar {X}}_{n}}$ is close to the normal distribution with mean ${\textstyle \mu }$ and variance ${\textstyle \sigma ^{2}/n}$. The usefulness of the theorem is that the distribution of ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ approaches normality regardless of the shape of the distribution of the individual ${\textstyle X_{i}}$. Formally, the theorem can be stated as follows:

Lindeberg–Lévy CLT. Suppose ${\textstyle \{X_{1},\ldots ,X_{n}\}}$ is a sequence of i.i.d. random variables with ${\textstyle \operatorname {E} [X_{i}]=\mu }$ and ${\textstyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty }$. Then as ${\textstyle n}$ approaches infinity, the random variables ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ converge in distribution to a normal ${\textstyle N(0,\sigma ^{2})}$:[3]

${\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\ \xrightarrow {d} \ N\left(0,\sigma ^{2}\right).}$

In the case ${\textstyle \sigma >0}$, convergence in distribution means that the cumulative distribution functions of ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ converge pointwise to the cdf of the ${\textstyle N(0,\sigma ^{2})}$ distribution: for every real number ${\textstyle z}$,

${\displaystyle \lim _{n\to \infty }\Pr \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\Pr \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\Phi \left({\frac {z}{\sigma }}\right),}$

where ${\textstyle \Phi (z)}$ is the standard normal cdf evaluated at ${\textstyle z}$. The convergence is uniform in ${\textstyle z}$ in the sense that

${\displaystyle \lim _{n\to \infty }\sup _{z\in \mathbb {R} }\left|\Pr \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0,}$

where ${\textstyle \mathrm {sup} }$ 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 ${\textstyle X_{i}}$ have to be independent, but not necessarily identically distributed. The theorem also requires that random variables ${\textstyle \vert X_{i}\vert }$ have moments of some order ${\textstyle (2+\delta )}$, and that the rate of growth of these moments is limited by the Lyapunov condition given below.

Lyapunov CLT.[5] Suppose ${\textstyle \{X_{1},\ldots ,X_{n}\}}$ is a sequence of independent random variables, each with finite expected value ${\textstyle \mu _{i}}$ and variance ${\textstyle \sigma _{i}^{2}}$. Define

${\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}}$

If for some ${\textstyle \delta >0}$, Lyapunov’s condition

${\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2+\delta }}}\sum _{i=1}^{n}\operatorname {E} \left[|X_{i}-\mu _{i}|^{2+\delta }\right]=0}$

is satisfied, then a sum of ${\textstyle {\frac {X_{i}-\mu _{i}}{s_{n}}}}$converges in distribution to a standard normal random variable, as ${\textstyle n}$ goes to infinity:

${\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)\ {\xrightarrow {d}}\ N(0,1).}$

In practice it is usually easiest to check Lyapunov's condition for ${\textstyle \delta =1}$.

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

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 ${\textstyle \epsilon >0}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\operatorname {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\{|X_{i}-\mu _{i}|>\varepsilon s_{n}\}}\right]=0}$

where ${\textstyle \mathbf {1} _{\{\ldots \}}}$ is the indicator function. Then the distribution of the standardized sums

${\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}$

converges towards the standard normal distribution ${\textstyle N(0,1)}$.

### Multidimensional CLT

Proofs that use characteristic functions can be extended to cases where each individual ${\textstyle \mathbf {X} _{i}}$ is a random vector in ${\textstyle \mathbb {R} ^{k}}$, with mean vector ${\textstyle \mu =\operatorname {E} [\mathbf {X} _{i}]}$ and covariance matrix ${\textstyle \mathbf {\Sigma } }$ (among the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done component-wise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.[6]

Let

${\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i(1)}\\\vdots \\X_{i(k)}\end{bmatrix}}}$

be the k-vector. The bold in ${\textstyle \mathbf {X} _{i}}$ means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be

${\displaystyle {\begin{bmatrix}X_{1(1)}\\\vdots \\X_{1(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2(1)}\\\vdots \\X_{2(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n(1)}\\\vdots \\X_{n(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}\left[X_{i(1)}\right]\\\vdots \\\sum _{i=1}^{n}\left[X_{i(k)}\right]\end{bmatrix}}=\sum _{i=1}^{n}\mathbf {X} _{i}}$

and the average is

${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}={\frac {1}{n}}{\begin{bmatrix}\sum _{i=1}^{n}X_{i(1)}\\\vdots \\\sum _{i=1}^{n}X_{i(k)}\end{bmatrix}}={\begin{bmatrix}{\bar {X}}_{i(1)}\\\vdots \\{\bar {X}}_{i(k)}\end{bmatrix}}=\mathbf {{\bar {X}}_{n}} }$

and therefore

${\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\operatorname {E} \left(X_{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right).}$

The multivariate central limit theorem states that

${\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\ {\stackrel {D}{\rightarrow }}\ N_{k}(0,{\boldsymbol {\Sigma }})}$

where the covariance matrix ${\displaystyle \mathbf {\Sigma } }$ is equal to

${\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\operatorname {Var} \left(X_{1(1)}\right)}&\operatorname {Cov} \left(X_{1(1)},X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(1)},X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(1)},X_{1(k)}\right)\\\operatorname {Cov} \left(X_{1(2)},X_{1(1)}\right)&\operatorname {Var} \left(X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(2)},X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(2)},X_{1(k)}\right)\\\operatorname {Cov} \left(X_{1(3)},X_{1(1)}\right)&\operatorname {Cov} \left(X_{1(3)},X_{1(2)}\right)&\operatorname {Var} \left(X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(3)},X_{1(k)}\right)\\\vdots &\vdots &\vdots &\ddots &\vdots \\\operatorname {Cov} \left(X_{1(k)},X_{1(1)}\right)&\operatorname {Cov} \left(X_{1(k)},X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(k)},X_{1(3)}\right)&\cdots &\operatorname {Var} \left(X_{1(k)}\right)\\\end{bmatrix}}.}$

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

Theorem.[7] Let ${\displaystyle X_{1},\dots ,X_{n}}$ be independent ${\displaystyle R^{d}}$-valued random vectors, each having mean zero. Write ${\displaystyle S=\sum _{i=1}^{n}X_{i}}$ and assume ${\displaystyle \Sigma =\operatorname {Cov} [S]}$ is invertible. Let ${\displaystyle Z\sim N(0,\Sigma )}$ be a ${\displaystyle d}$-dimensional Gaussian with the same mean and covariance matrix as ${\displaystyle S}$. Then for all convex sets ${\displaystyle U\subseteq R^{d}}$,

${\displaystyle |\Pr[S\in U]-\Pr[Z\in U]|\leq Cd^{1/4}\gamma ,}$

where ${\displaystyle C}$ is a universal constant, ${\displaystyle \gamma =\sum _{i=1}^{n}\operatorname {E} [\|\Sigma ^{-1/2}X_{i}\|_{2}^{3}]}$, and ${\displaystyle \|\cdot \|_{2}}$ denotes the Euclidean norm on ${\displaystyle R^{d}}$.

It is unknown whether the factor ${\textstyle d^{1/4}}$ 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 power-law tail (Paretian tail) distributions decreasing as ${\textstyle {\vert x\vert }^{-a-1}}$ where ${\textstyle 0<\alpha <2}$ (and therefore having infinite variance) will tend to a stable distribution ${\textstyle f(x;\alpha ,0,c,0)}$ as the number of summands grows.[9][10] If ${\textstyle \alpha >2}$ 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 α-mixing) defined by ${\textstyle \alpha (n)\to 0}$ where ${\textstyle \alpha (n)}$ is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is:[12]

Theorem. Suppose that ${\textstyle \{X_{1},\ldots ,X_{n}\}}$ is stationary and ${\displaystyle \alpha }$-mixing with ${\textstyle \alpha _{n}=O(n^{-5})}$ and that ${\textstyle \operatorname {E} [X_{n}]=0}$ and ${\textstyle \operatorname {E} [{X_{n}}^{12}]<\infty }$. Denote ${\textstyle S_{n}=X_{1}+\cdots +X_{n}}$, then the limit

${\displaystyle \sigma ^{2}=\lim _{n}{\frac {\operatorname {E} \left(S_{n}^{2}\right)}{n}}}$

exists, and if ${\textstyle \sigma \neq 0}$ then ${\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}}$ converges in distribution to ${\textstyle N(0,1)}$.

In fact,

${\displaystyle \sigma ^{2}=\operatorname {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\operatorname {E} \left(X_{1}X_{1+k}\right),}$

where the series converges absolutely.

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

There is a stronger version of the theorem:[13] the assumption ${\textstyle \operatorname {E} [{X_{n}}^{12}]<\infty }$ is replaced with ${\textstyle \operatorname {E} [{\vert X_{n}\vert }^{2+\delta }]<\infty }$, and the assumption ${\textstyle \alpha _{n}=O(n^{-5})}$ is replaced with

${\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta )}}<\infty .}$

Existence of such ${\textstyle \delta >0}$ ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).

### Martingale difference CLT

Theorem. Let a martingale ${\textstyle M_{n}}$ satisfy

• ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left(\left(M_{k}-M_{k-1}\right)^{2}|M_{1},\dots ,M_{k-1}\right)\to 1}$ in probability as n → ∞,
• for every ε > 0, ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left(\left(M_{k}-M_{k-1}\right)^{2};|M_{k}-M_{k-1}|>\varepsilon {\sqrt {n}}\right)\to 0}$ as n → ∞,

then ${\textstyle {\frac {M_{n}}{\sqrt {n}}}}$ converges in distribution to ${\textstyle N(0,1)}$ as ${\textstyle n\to \infty }$.[14][15]

Caution: The restricted expectation[clarification needed] ${\textstyle \operatorname {E} [X;A]}$ should not be confused with the conditional expectation ${\textstyle \operatorname {E} [X\mid A]={\frac {\operatorname {E} [X;A]}{\mathbf {P} (A)}}}$.

## 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 ${\textstyle \{X_{1},\ldots ,X_{n}\}}$ are independent and identically distributed random variables, each with mean ${\textstyle \mu }$ and finite variance ${\textstyle \sigma ^{2}}$. The sum ${\textstyle X_{1}+\cdots +X_{n}}$ has mean ${\textstyle n\mu }$ and variance ${\textstyle n\sigma ^{2}}$. Consider the random variable

${\displaystyle Z_{n}\ =\ {\frac {X_{1}+\cdots +X_{n}-n\mu }{\sqrt {n\sigma ^{2}}}}\ =\ \sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sqrt {n\sigma ^{2}}}}\ =\ \sum _{i=1}^{n}{\frac {1}{\sqrt {n}}}Y_{i},}$

where in the last step we defined the new random variables ${\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}}$, each with zero mean and unit variance (${\textstyle \operatorname {var} (Y)=1}$). The characteristic function of ${\textstyle Z_{n}}$ is given by

${\displaystyle \varphi _{Z_{n}}\!(t)\ =\ \varphi _{\sum _{i=1}^{n}{{\frac {1}{\sqrt {n}}}Y_{i}}}\!(t)\ =\ \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\varphi _{Y_{2}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\cdots \varphi _{Y_{n}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ \left[\varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\right]^{n},}$

where in the last step we used the fact that all of the ${\textstyle Y_{i}}$ are identically distributed. The characteristic function of ${\textstyle Y_{1}}$ is, by Taylor's theorem,

${\displaystyle \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ 1-{\frac {t^{2}}{2n}}+o\!\!\left({\frac {t^{2}}{n}}\right),\quad {\bigg (}{\frac {t}{\sqrt {n}}}{\bigg )}\rightarrow 0}$

where ${\textstyle o(t^{2}/n)}$ is "little o notation" for some function of ${\textstyle t}$ that goes to zero more rapidly than ${\textstyle t^{2}/n}$. By the limit of the exponential function (${\textstyle e^{x}=\lim {(1+{\frac {x}{n}})^{n}}}$), the characteristic function of ${\displaystyle Z_{n}}$ equals

${\displaystyle \varphi _{Z_{n}}(t)=\left(1-{\frac {t^{2}}{2n}}+o\left({\frac {t^{2}}{n}}\right)\right)^{n}\rightarrow e^{-{\frac {1}{2}}t^{2}},\quad n\rightarrow \infty .}$

All of the higher order terms vanish in the limit ${\textstyle n\to \infty }$. The right hand side equals the characteristic function of a standard normal distribution ${\textstyle N(0,1)}$, which implies through Lévy's continuity theorem that the distribution of ${\textstyle Z_{n}}$ will approach ${\textstyle N(0,1)}$ as ${\textstyle n\to \infty }$. Therefore, the sum ${\textstyle X_{1}+\cdots +X_{n}}$ will approach that of the normal distribution ${\textstyle N(n\mu ,n\sigma ^{2})}$, and the sample average

${\displaystyle {\bar {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}$

converges to the normal distribution ${\textstyle N(\mu ,{\frac {\sigma ^{2}}{n}})}$, 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 ${\textstyle \operatorname {E} [(X_{1}-\mu )^{3}]}$ exists and is finite, then the speed of convergence is at least on the order of ${\textstyle 1/{\sqrt {n}}}$ (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 ${\textstyle Z_{n}}$ 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 Sn 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 ${\textstyle f(n)}$:

${\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O{\big (}\varphi _{3}(n){\big )}\qquad (n\rightarrow \infty ).}$

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.

${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}$

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):

${\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}.}$

Here one can say that the difference between the function and its approximation grows approximately as 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.[20] If in addition each Xi has finite variance σ2, then by the central limit theorem,

${\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\rightarrow \xi ,}$

where ξ is distributed as N(0,σ2). This provides values of the first two constants in the informal expansion

${\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}$

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:

${\displaystyle {\frac {S_{n}-a_{n}}{b_{n}}}\rightarrow \Xi ,}$

or informally

${\displaystyle S_{n}\approx a_{n}+\Xi b_{n}.}$

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 bn may be proportional to nc, for any c1/2; 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 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. 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 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 σ2 and function f(n) such that Sn/(σn⋅f(n)) converges in distribution to N(0,1) as n→ ∞.

Lemma.[24] Suppose ${\displaystyle X_{1},X_{2},\dots }$ is a sequence of real-valued and strictly stationary random variables with ${\displaystyle \mathbb {E} (X_{i})=0}$ for all ${\displaystyle i}$, ${\displaystyle g:[0,1]\rightarrow \mathbb {R} }$, and ${\displaystyle S_{n}=\sum _{i=1}^{n}g({\tfrac {i}{n}})X_{i}}$. Construct

${\displaystyle \sigma ^{2}=\mathbb {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})}$
1. If ${\displaystyle \sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})}$ is absolutely convergent, ${\displaystyle \left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty }$, and ${\displaystyle 0<\int _{0}^{1}(g(x))^{2}dx<\infty }$ then ${\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\rightarrow \sigma ^{2}}$ as ${\displaystyle n\rightarrow \infty }$ where ${\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}(g({\tfrac {i}{n}}))^{2}}$.
2. If in addition ${\displaystyle \sigma >0}$ and ${\displaystyle S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}}$ converges in distribution to ${\displaystyle {\mathcal {N}}(0,1)}$ as ${\displaystyle n\rightarrow \infty }$ then ${\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}$ also converges in distribution to ${\displaystyle {\mathcal {N}}(0,1)}$ as ${\displaystyle n\rightarrow \infty }$.

## 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 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.[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 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(X2
k
) = 1
for all k = 1, …, n. Then the distribution of

${\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}}$

is εn-close to N(0,1) in the total variation distance.[26]

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.[27]

Here is a Berry–Esseen type result.

Theorem. Let X1, …, Xn satisfy the assumptions of the previous theorem, then [28]

${\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}\mathrm {e} ^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}}$

for all a < b; here C is a universal (absolute) constant. Moreover, for every c1, …, cn such that c2
1
+ … + c2
n
= 1
,

${\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}\mathrm {e} ^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}$

The distribution of X1 + … + Xn/n need not be approximately normal (in fact, it can be uniform).[29] However, the distribution of c1X1 + … + cnXn is close to N(0,1) (in the total variation distance) for most vectors (c1, …, cn) according to the uniform distribution on the sphere c2
1
+ … + c2
n
= 1
.

### Lacunary trigonometric series

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

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

Then[30][31]

${\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}}$

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

### Gaussian polytopes

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[32]

${\displaystyle {\frac {X_{n}-\mathrm {E} (X_{n})}{\sqrt {\operatorname {Var} (X_{n})}}}}$

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

The same 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.[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,); 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] 23/n − 1.

### Subsequences

Theorem. Let random variables X1, X2, … ∈ L2(Ω) be such that Xn → 0 weakly in L2(Ω) and X
n
→ 1
weakly in L1(Ω). Then there exist integers n1 < n2 < … such that

${\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}}$

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 infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures. [36][37]

## Applications and examples

### Simple example

This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the chi-squared values that quantify the goodness of the fit (the fit is good if the reduced chi-squared value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~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.

Comparison of probability density functions, **p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 512. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.

### 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 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

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

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 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:[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 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.[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 = ex2 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.[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]

## Notes

1. ^ Montgomery, Douglas C.; Runger, George C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley. p. 241. ISBN 9781118539712.
2. ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 10.
3. ^ Billingsley (1995, p. 357)
4. ^ Bauer (2001, Theorem 30.13, p.199)
5. ^ Billingsley (1995, p.362)
6. ^ Van der Vaart, A. W. (1998). Asymptotic statistics. New York: Cambridge University Press. ISBN 978-0-521-49603-2. LCCN 98015176.
7. ^ Ryan O’Donnell (2014, Theorem 5.38) http://www.contrib.andrew.cmu.edu/~ryanod/?p=866
8. ^ Bentkus, V. (2005). "A Lyapunov-type Bound in ${\displaystyle \mathbb {R} ^{d}}$". Theory Probab. Appl. 49 (2): 311–323. doi:10.1137/S0040585X97981123.
9. ^ Voit, Johannes (2003). "Section 5.4.3". The Statistical Mechanics of Financial Markets. Texts and Monographs in Physics. Springer-Verlag. ISBN 3-540-00978-7.
10. ^ Gnedenko, B. V.; Kolmogorov, A. N. (1954). Limit distributions for sums of independent random variables. Cambridge: Addison-Wesley.
11. ^ a b Uchaikin, Vladimir V.; Zolotarev, V. M. (1999). Chance and stability: stable distributions and their applications. VSP. pp. 61–62. ISBN 90-6764-301-7.
12. ^ Billingsley (1995, Theorem 27.5)
13. ^ Durrett (2004, Sect. 7.7(c), Theorem 7.8)
14. ^ Durrett (2004, Sect. 7.7, Theorem 7.4)
15. ^ Billingsley (1995, Theorem 35.12)
16. ^ "An Introduction to Stochastic Processes in Physics". jhupbooks.press.jhu.edu. Retrieved 2016-08-11.
17. ^ 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.
18. ^ Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). Normal approximation by Stein's method. Springer. ISBN 978-3-642-15006-7.
19. ^ 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/S0894-0347-04-00459-X
20. ^ Rosenthal, Jeffrey Seth (2000). A First Look at Rigorous Probability Theory. World Scientific. Theorem 5.3.4, p. 47. ISBN 981-02-4322-7.
21. ^ Johnson, Oliver Thomas (2004). Information Theory and the Central Limit Theorem. Imperial College Press. p. 88. ISBN 1-86094-473-6.
22. ^ 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 0-8218-0438-3.
23. ^ Petrov, V. V. (1976). Sums of Independent Random Variables. New York-Heidelberg: Springer-Verlag. ch. 7. ISBN 9783642658099.
24. ^ 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.
25. ^ Rempala, G.; Wesolowski, J. (2002). "Asymptotics of products of sums and U-statistics" (PDF). Electronic Communications in Probability. 7: 47–54. doi:10.1214/ecp.v7-1046.
26. ^ Klartag (2007, Theorem 1.2)
27. ^ Durrett (2004, Section 2.4, Example 4.5)
28. ^ Klartag (2008, Theorem 1)
29. ^ Klartag (2007, Theorem 1.1)
30. ^ Zygmund, Antoni (2003) [1959]. Trigonometric Series. Cambridge University Press. vol. II, sect. XVI.5, Theorem 5-5. ISBN 0-521-89053-5.
31. ^ Gaposhkin (1966, Theorem 2.1.13)
32. ^ Bárány & Vu (2007, Theorem 1.1)
33. ^ Bárány & Vu (2007, Theorem 1.2)
34. ^ 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/S0002-9947-08-04444-9.
35. ^ Gaposhkin (1966, Sect. 1.5)
36. ^ Kotani, M.; Sunada, Toshikazu (2003). Spectral geometry of crystal lattices. 338. Contemporary Math. pp. 271–305. ISBN 978-0-8218-4269-0.
37. ^ Sunada, Toshikazu (2012). Topological Crystallography – With a View Towards Discrete Geometric Analysis. Surveys and Tutorials in the Applied Mathematical Sciences. 6. Springer. ISBN 978-4-431-54177-6.
38. ^ Dinov, Christou & Sánchez (2008)
39. ^ "SOCR EduMaterials Activities GCLT Applications - Socr". Wiki.stat.ucla.edu. 2010-05-24. Retrieved 2017-01-23.
40. ^ 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. Cite journal requires |journal= (help)
41. ^ Henk, Tijms (2004). Understanding Probability: Chance Rules in Everyday Life. Cambridge: Cambridge University Press. p. 169. ISBN 0-521-54036-4.
42. ^ Galton, F. (1889). Natural Inheritance. p. 66.
43. ^ 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.
44. ^ a b c Le Cam, Lucien (1986). "The central limit theorem around 1935". Statistical Science. 1 (1): 78–91. doi:10.1214/ss/1177013818.
45. ^ Hald, Andreas (22 April 1998). A History of Mathematical Statistics from 1750 to 1930 (PDF). Gbv.de. chapter 17. ISBN 978-0471179122.
46. ^ 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:10.1007/978-0-387-87857-7. ISBN 978-0-387-87856-0. 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)
47. ^ 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.
48. ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 977. ISBN 1-57955-008-8.
49. ^ Hodges, Andrew (1983). Alan Turing: The Enigma. London: Burnett Books. pp. 87–88. ISBN 0091521300.
50. ^ Zabell, S. L. (2005). Symmetry and Its Discontents: Essays on the History of Inductive Probability. Cambridge University Press. p. 199. ISBN 0-521-44470-5.
51. ^ Aldrich, John (2009). "England and Continental Probability in the Inter-War Years". Electronic Journ@l for History of Probability and Statistics. 5 (2). Section 3.
52. ^ Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall. ISBN 978-0412997112.