Central limit theorem

In probability theory, the central limit theorem states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.
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 form it was only precisely stated in the 1920s.
In statistics, the CLT can be stated as: let denote a statistical sample of size from a population with expected value and finite positive variance, and let denote the sample mean. Then the limit as of the distribution of is a normal distribution with mean and variance.
In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the probability distribution of these averages will closely approximate a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed. This requirement can be weakened; 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

Classical CLT

Let be a sequence of i.i.d. random variables having a distribution with expected value given by and finite variance given by Suppose we are interested in the sample average
By the law of large numbers, the sample average converges almost surely to the expected value as
The classical central limit theorem describes the size and the distributional form of the fluctuations around the deterministic number during this convergence. More precisely, it states that as gets larger, the distribution of the normalized mean, i.e. the difference between the sample average and its limit scaled by the factor, approaches the normal distribution with mean and variance For large enough the distribution of gets arbitrarily close to the normal distribution with mean and variance
The usefulness of the theorem is that the distribution of approaches normality regardless of the shape of the distribution of the individual Formally, the theorem can be stated as follows:
In the case convergence in distribution means that the cumulative distribution functions of converge pointwise to the cdf of the distribution: for every real number
where is the standard normal cdf evaluated at The convergence is uniform in in the sense that
where denotes the least upper bound of the set.

Lyapunov CLT

In this variant of the central limit theorem the random variables have to be independent, but not necessarily identically distributed. The theorem also requires that random variables have moments of some order and that the rate of growth of these moments is limited by the Lyapunov condition given below.
In practice it is usually easiest to check Lyapunov's condition for
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 (-Feller) CLT

In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one.
Suppose that for every,
where is the indicator function. Then the distribution of the standardized sums
converges towards the standard normal distribution

CLT for the sum of a random number of random variables

Rather than summing an integer number of random variables and taking, the sum can be of a random number of random variables, with conditions on. For example, the following theorem is Corollary 4 of Robbins. It assumes that is asymptotically normal.

Multidimensional CLT

Proofs that use characteristic functions can be extended to cases where each individual is a random vector in with mean vector and covariance matrix , and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution. Summation of these vectors is done component-wise.
For let
be independent random vectors. The sum of the random vectors is
and their average is
Therefore,
The multivariate central limit theorem states that
where the covariance matrix is equal to
The multivariate central limit theorem can be proved using the Cramér–Wold theorem.
The rate of convergence is given by the following Berry–Esseen type result:
It is unknown whether the factor is necessary.

The generalized central limit theorem

The generalized central limit theorem was an effort of multiple mathematicians over the period from 1920 to 1937. The first published complete proof of the GCLT was in 1937 by Paul Lévy in French. An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.
The statement of the GCLT is as follows:
In other words, if sums of independent, identically distributed random variables converge in distribution to some, then must be a stable distribution.

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 defined by where is so-called strong mixing coefficient.
A simplified formulation of the central limit theorem under strong mixing is:
In fact,
where the series converges absolutely.
The assumption cannot be omitted, since the asymptotic normality fails for where are another stationary sequence.
There is a stronger version of the theorem: the assumption is replaced with and the assumption is replaced with
Existence of such ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see.

Martingale difference CLT

Remarks

Proof of classical CLT

The central limit theorem has a proof using characteristic functions. It is similar to the proof of the law of large numbers.
Assume are independent and identically distributed random variables, each with mean and finite variance The sum has mean and variance Consider the random variable
where in the last step we defined the new random variables each with zero mean and unit variance The characteristic function of is given by
where in the last step we used the fact that all of the are identically distributed. The characteristic function of is, by Taylor's theorem,
where is "little notation" for some function of that goes to zero more rapidly than By the limit of the exponential function the characteristic function of equals
All of the higher order terms vanish in the limit The right hand side equals the characteristic function of a standard normal distribution, which implies through Lévy's continuity theorem that the distribution of will approach as Therefore, the sample average
is such that
converges to the normal distribution 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 exists and is finite, then the speed of convergence is at least on the order of . Stein's method can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.
The convergence to the normal distribution is monotonic, in the sense that the entropy of increases monotonically to that of the normal distribution.
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. This means that if we build a histogram of the realizations of the sum of 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 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.