Classical CLT

Let ${\displaystyle \{X_{1},\ldots ,X_{n}}\}$ be a sequence of i.i.d. random variables having a distribution with expected value given by ${\displaystyle \mu }$ and finite variance given by ${\displaystyle \sigma ^{2}.}$ Suppose we are interested in the sample average

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

By the law of large numbers, the sample average converges almost surely (and therefore also converges in probability) to the expected value ${\displaystyle \mu }$ as ${\displaystyle n\to \infty .}$

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number ${\displaystyle \mu }$ during this convergence. More precisely, it states that as ${\displaystyle n}$ gets larger, the distribution of the difference between the sample average ${\displaystyle {\bar {X}}_{n}}$ and its limit ${\displaystyle \mu ,}$ when multiplied by the factor ${\displaystyle {\sqrt {n}}}$ (that is, ${\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$) approximates the normal distribution with mean ${\displaystyle 0}$ and variance ${\displaystyle \sigma ^{2}.}$ For large enough ${\displaystyle n,}$ the distribution of ${\displaystyle {\bar {X}}_{n}}$ gets arbitrarily close to the normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}/n.}$

The usefulness of the theorem is that the distribution of ${\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ approaches normality regardless of the shape of the distribution of the individual ${\displaystyle X_{i}.}$ Formally, the theorem can be stated as follows:

Lindeberg–Lévy CLT — Suppose ${\displaystyle X_{1},X_{2},X_{3}\ldots }$ is a sequence of i.i.d. random variables with ${\displaystyle \operatorname {E} [X_{i}]=\mu }$ and ${\displaystyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty .}$ Then, as ${\displaystyle n}$ approaches infinity, the random variables ${\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ converge in distribution to a normal ${\displaystyle {\mathcal {N}}(0,\sigma ^{2})}$:^[4]

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

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

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

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

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

where ${\displaystyle \sup }$ denotes the least upper bound (or supremum) of the set.^[5]

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 ${\textstyle \left|X_{i}\right|}$ 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^[6] — Suppose ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ 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

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

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:

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

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 ${\textstyle \varepsilon >0}$

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

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 ${\textstyle {\mathcal {N}}(0,1)}$.

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 ${\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]}$ 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.^[7]

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

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

The multivariate central limit theorem states that

$${\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\,\xrightarrow {D} \ {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }})}$$

where the covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$ 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^[8] — Let ${\displaystyle X_{1},\dots ,X_{n},\dots }$ be independent ${\displaystyle \mathbb {R} ^{d}}$-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 ${\displaystyle Z\sim {\mathcal {N}}(0,\Sigma )}$ be a ${\displaystyle d}$-dimensional Gaussian with the same mean and same covariance matrix as ${\displaystyle S}$. Then for all convex sets ${\displaystyle U\subseteq \mathbb {R} ^{d}}$,

$${\displaystyle \left|\mathbb {P} [S\in U]-\mathbb {P} [Z\in U]\right|\leq C\,d^{1/4}\gamma ~,}$$

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

It is unknown whether the factor ${\textstyle d^{1/4}}$ is necessary.^[9]

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:^[14]

Theorem — Suppose that ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ is stationary and ${\displaystyle \alpha }$-mixing with ${\textstyle \alpha _{n}=O\left(n^{-5}\right)}$ 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

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

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

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:^[15] the assumption ${\textstyle \operatorname {E} \left[X_{n}^{12}\right]<\infty }$ is replaced with ${\textstyle \operatorname {E} \left[{\left|X_{n}\right|}^{2+\delta }\right]<\infty }$, and the assumption ${\textstyle \alpha _{n}=O\left(n^{-5}\right)}$ 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

Proof of classical CLT

The central limit theorem has a proof using characteristic functions.^[18] It is similar to the proof of the (weak) law of large numbers.

Assume ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ 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,

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

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 (${\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}$), the characteristic function of ${\displaystyle Z_{n}}$ equals

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

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 ${\textstyle {\mathcal {N}}(0,1)}$, which implies through Lévy's continuity theorem that the distribution of ${\textstyle Z_{n}}$ will approach ${\textstyle {\mathcal {N}}(0,1)}$ as ${\textstyle n\to \infty }$. Therefore, the sample average

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

is such that

$${\displaystyle {\frac {\sqrt {n}}{\sigma }}({\bar {X}}_{n}-\mu )=Z_{n}}$$

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

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment ${\textstyle \operatorname {E} \left[(X_{1}-\mu )^{3}\right]}$ 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^[19] can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.^[20]

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.^[21]

The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realizations of the sum of n independent identical discrete variables, the piecewise-linear 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.

Common misconceptions

Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely-used textbooks.^[22][23][24] These include the beliefs that:

• The theorem applies to random sampling of any variable, rather than to the mean values (or sums) of iid random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables.
• The theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the Glivenko-Cantelli theorem.
• That the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,^[25] allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences.

Relation to the law of large numbers

The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of S_n as n approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of ${\textstyle f(n)}$:

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

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, S_n, of independent identically distributed random variables, X₁, ..., X_n, is studied in classical probability theory. If each X_i has finite mean μ, then by the law of large numbers, S_n/n → μ.^[26] If in addition each X_i has finite variance σ², then by the central limit theorem,

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

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 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:

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

or informally

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

Distributions Ξ which can arise in this way are called stable.^[27] 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.^[28][29]

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

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 σ² and function f(n) such that S_n/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.

Lemma^[31] — Suppose ${\displaystyle X_{1},X_{2},\dots }$ is a sequence of real-valued and strictly stationary random variables with ${\displaystyle \operatorname {E} (X_{i})=0}$ for all ${\displaystyle i}$, ${\displaystyle g:[0,1]\to \mathbb {R} }$, and ${\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}$. Construct

$${\displaystyle \sigma ^{2}=\operatorname {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}$$

1. If ${\displaystyle \sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}$ 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 ${\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\to \sigma ^{2}}$ as ${\displaystyle n\to \infty }$ where ${\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}\left(g\left({\tfrac {i}{n}}\right)\right)^{2}}$.
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 ${\displaystyle n\to \infty }$ then ${\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}$ also converges in distribution to ${\displaystyle {\mathcal {N}}(0,1)}$ as ${\displaystyle n\to \infty }$.

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.^[32]

Convex body

Theorem — There exists a sequence ε_n ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X₁, ..., X_n have a log-concave joint density f such that f(x₁, ..., x_n) = f(|x₁|, ..., |x_n|) for all x₁, ..., x_n, and E(X²
_k) = 1 for all k = 1, ..., n. Then the distribution of

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

is ε_n-close to ${\textstyle {\mathcal {N}}(0,1)}$ in the total variation distance.^[33]

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

The condition f(x₁, ..., x_n) = f(|x₁|, ..., |x_n|) ensures that X₁, ..., X_n are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem.^[34]

Here is a Berry–Esseen type result.

Theorem — Let X₁, ..., X_n satisfy the assumptions of the previous theorem, then ^[35]

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

for all a < b; here C is a universal (absolute) constant. Moreover, for every c₁, ..., c_n ∈ R such that c²
₁ + ⋯ + c²
_n = 1,

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

The distribution of X₁ + ⋯ + X_n/√n need not be approximately normal (in fact, it can be uniform).^[36] However, the distribution of c₁X₁ + ⋯ + c_nX_n is close to ${\textstyle {\mathcal {N}}(0,1)}$ (in the total variation distance) for most vectors (c₁, ..., c_n) according to the uniform distribution on the sphere c²
₁ + ⋯ + c²
_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_kU + 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
  $${\displaystyle r_{1}^{2}+r_{2}^{2}+\cdots =\infty \quad {\text{ and }}\quad {\frac {r_{k}^{2}}{r_{1}^{2}+\cdots +r_{k}^{2}}}\to 0,}$$

  
• 0 ≤ a_k < 2π.

Then^[37][38]

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

converges in distribution to ${\textstyle {\mathcal {N}}{\big (}0,{\frac {1}{2}}{\big )}}$.

Gaussian polytopes

Theorem — Let A₁, ..., A_n be independent random points on the plane R² each having the two-dimensional standard normal distribution. Let K_n be the convex hull of these points, and X_n the area of K_n Then^[39]

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

converges in distribution to ${\textstyle {\mathcal {N}}(0,1)}$ 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.^[40]

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.

Subsequences

Theorem — Let random variables X₁, X₂, ... ∈ L₂(Ω) be such that X_n → 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 ${\textstyle {\mathcal {N}}(0,1)}$ as k tends to infinity.^[42]

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.^[43][44]

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.^[45]