Central limit theorem

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.

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920,[1] thereby serving as a bridge between classical and modern probability theory.

If {textstyle X_{1},X_{2},dots ,X_{n},dots } are random samples drawn from a population with overall mean {textstyle mu } and finite variance {textstyle sigma ^{2}} , and if {textstyle {bar {X}}_{n}} is the sample mean of the first {textstyle n} samples, then the limiting form of the distribution, {textstyle Z=lim _{nto infty }{left({frac {{bar {X}}_{n}-mu }{sigma _{bar {X}}}}right)}} , with {displaystyle sigma _{bar {X}}={frac {sigma }{sqrt {n}}}} , is a standard normal distribution.[2] 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.

Contents 1 Independent sequences 1.1 Classical CLT 1.2 Lyapunov CLT 1.3 Lindeberg CLT 1.4 Multidimensional CLT 1.5 Generalized theorem 2 Dependent processes 2.1 CLT under weak dependence 2.2 Martingale difference CLT 3 Remarks 3.1 Proof of classical CLT 3.2 Convergence to the limit 3.3 Relation to the law of large numbers 3.4 Alternative statements of the theorem 3.4.1 Density functions 3.4.2 Characteristic functions 3.5 Calculating the variance 4 Extensions 4.1 Products of positive random variables 5 Beyond the classical framework 5.1 Convex body 5.2 Lacunary trigonometric series 5.3 Gaussian polytopes 5.4 Linear functions of orthogonal matrices 5.5 Subsequences 5.6 Random walk on a crystal lattice 6 Applications and examples 6.1 Simple example 6.2 Real applications 7 Regression 7.1 Other illustrations 8 History 9 See also 10 Notes 11 References 12 External links 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.[3] Classical CLT Let {textstyle {X_{1},ldots ,X_{n},ldots }} be a sequence of random samples — 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}equiv {frac {X_{1}+cdots +X_{n}}{n}}} of the first {textstyle n} samples.

By the law of large numbers, the sample averages converge almost surely (and therefore also converge in probability) to the expected value {textstyle mu } as {textstyle nto 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},ldots }} is a sequence of i.i.d. random variables with {textstyle mathbb {E} [X_{i}]=mu } and {textstyle operatorname {Var} [X_{i}]=sigma ^{2}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 {mathcal {N}}(0,sigma ^{2})} distribution: for every real number {textstyle z} , {displaystyle lim _{nto infty }mathbb {P} left[{sqrt {n}}({bar {X}}_{n}-mu )leq zright]=lim _{nto infty }mathbb {P} 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 _{nto infty };sup _{zin mathbb {R} };left|mathbb {P} left[{sqrt {n}}({bar {X}}_{n}-mu )leq zright]-Phi left({frac {z}{sigma }}right)right|=0~,} where {textstyle 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 _{nto infty };{frac {1}{s_{n}^{2+delta }}},sum _{i=1}^{n}mathbb {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 Main article: Lindeberg's condition 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 _{nto infty }{frac {1}{s_{n}^{2}}}sum _{i=1}^{n}mathbb {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 }}=mathbb {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}-mathbb {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),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 Zsim {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 Usubseteq mathbb {R} ^{d}} , {displaystyle left|mathbb {P} [Sin U]-mathbb {P} [Zin U]right|leq C,d^{1/4}gamma ~,} where {displaystyle C} is a universal constant, {displaystyle gamma =sum _{i=1}^{n}mathbb {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] Generalized theorem Main article: Stable distribution § A generalized central limit 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 {|x|}^{-alpha -1}} where {textstyle 02} then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution.[12] 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:[13] Theorem — Suppose that {textstyle {X_{1},ldots ,X_{n},ldots }} is stationary and {displaystyle alpha } -mixing with {textstyle alpha _{n}=Oleft(n^{-5}right)} and that {textstyle mathbb {E} [X_{n}]=0} and {textstyle mathbb {E} [{X_{n}}^{12}]0} ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).

Martingale difference CLT Main article: Martingale central limit theorem Theorem — Let a martingale {textstyle M_{n}} satisfy {displaystyle {frac {1}{n}}sum _{k=1}^{n}mathbb {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}{mathbb {E} left[left(M_{k}-M_{k-1}right)^{2}mathbf {1} left[|M_{k}-M_{k-1}|>varepsilon {sqrt {n}}right]right]}to 0} as n → ∞, then {textstyle {frac {M_{n}}{sqrt {n}}}} converges in distribution to {textstyle N(0,1)} as {textstyle nto infty } .[15][16] Remarks Proof of classical CLT The central limit theorem has a proof using characteristic functions.[17] 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 nmu } and variance {textstyle nsigma ^{2}} . Consider the random variable {displaystyle Z_{n}={frac {X_{1}+cdots +X_{n}-nmu }{sqrt {nsigma ^{2}}}}=sum _{i=1}^{n}{frac {X_{i}-mu }{sqrt {nsigma ^{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 _{nto 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}}+oleft({frac {t^{2}}{n}}right)right)^{n}rightarrow e^{-{frac {1}{2}}t^{2}},quad nto infty .} All of the higher order terms vanish in the limit {textstyle nto 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 nto 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 )} converges to the normal distribution {textstyle 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.[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} 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[18] can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.[19] The convergence to the normal distribution is monotonic, in the sense that the entropy of {textstyle Z_{n}} increases monotonically to that of the normal distribution.[20] The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realizations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity, this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.

Relation to the law of large numbers 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 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 (nto 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 _{nto 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 _{nto 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 / n → μ.[21] If in addition each Xi has finite variance σ2, then by the central limit theorem, {displaystyle {frac {S_{n}-nmu }{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 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.[22] 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 c ≥ 1 / 2 ; it may also be multiplied by a slowly varying function of n.[12][23] 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[24] 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/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.

Lemma[25] — 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]to mathbb {R} } , and {displaystyle S_{n}=sum _{i=1}^{n}gleft({tfrac {i}{n}}right)X_{i}} . Construct {displaystyle sigma ^{2}=mathbb {E} (X_{1}^{2})+2sum _{i=1}^{infty }mathbb {E} (X_{1}X_{1+i})} 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),dxright|0} and {displaystyle S_{n}/{sqrt {mathrm {Var} (S_{n})}}} converges in distribution to {displaystyle {mathcal {N}}(0,1)} as {displaystyle nto infty } then {displaystyle S_{n}/(sigma {sqrt {ngamma _{n}}})} also converges in distribution to {displaystyle {mathcal {N}}(0,1)} as {displaystyle nto 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.[26] 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.[27] These two εn-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

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

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