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Central limit theorem
Central limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results explain the ubiquity of the normal distribution.
The most important and famous result is simply called The Central Limit Theorem; it is concerned with independent variables with identical distribution whose expected value and variance are finite. Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others. Two such conditions are the Lindeberg condition and the Lyapunov condition. Other generalizations even allow some "weak" dependence of the random variables.
The reader may find it helpful to consider this illustration of the central limit theorem.
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"The" central limit theorem
Let X_{1},X_{2},X_{3},... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the expected value μ and the standard deviation σ of D exist and are finite.
Consider the sum :S_{n}=X_{1}+...+X_{n}. Then the expected value of S_{n} is nμ and its standard deviation is σ n^{½}. Furthermore, informally speaking, the distribution of S_{n} approaches the normal distribution N(nμ,σ^{2}n) as n approaches ∞.
In order to clarify the word "approaches" in the last sentence, we standardize S_{n} by setting
Then the distribution of Z_{n} converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution). This means: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have
or, equivalently,
where
is the "sample mean".
Proof of the central limit theorem
For a theorem of such fundamental importance to statistics and applied probability, the central limit theorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (weak) law of large numbers. For any random variable, Y, with zero mean and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,
Letting Y_{i} be (X_{i} − μ)/σ, the standardised value of X_{i}, it is easy to see that the standardised mean of the observations X_{1}, X_{2}, ..., X_{n} is just
By simple properties of characteristic functions, the characteristic function of Z_{n} is
But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem , which confirms that the convergence of characteristic functions implies convergence in distribution.
Convergence to the limit
If the third central moment E((X_{1} − μ)^{3}) exists and is finite, then the above convergence is uniform and the speed of convergence is at least on the order of 1/n^{½} (see Berry-Esséen theorem).
Pictures of a distribution being "smoothed out" by summation (showing original distribution and three subsequent convolutions):
(See Illustration of the central limit theorem for further details on these images.)
An equivalent formulation of this limit theorem starts with A_{n} = (X_{1} + ... + X_{n}) / n which can be interpreted as the mean of a random sample of size n. The expected value of A_{n} is μ and the standard deviation is σ / n^{½}. If we normalize A_{n} by setting Z_{n} = (A_{n} - μ) / (σ / n^{½}), we obtain the same variable Z_{n} as above, and it approaches a standard normal distribution.
Note the following apparent "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives? The reason is simple: the theorem applies to terms centered about the mean. Without that standardization, the distribution would, as intuition suggests, escape away to infinity.
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, under the conditions stated above.
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 tends to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above.
An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.
Products of random variables
The central limit theorem tells us what to expect about the sum of independant random variables, but what about the product? Well, the logarithm of a product is simply the sum of the logs of the factors, so the log of a product of random variables tends to have a normal distribution, which makes the product itself have a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the product of different random factors, so they follow a log-normal distribution.
Lyapunov condition
Let X_{n} be a sequence of independent random variables defined on the same probability space. Assume that X_{n} has finite expected value μ_{n} and finite standard deviation σ_{n}. We define
Assume that the third central moments
are finite for every n, and that
(This is the Lyapunov condition). We again consider the sum S_{n}=X_{1}+...+X_{n}. The expected value of S_{n} is m_{n} = ∑_{i=1..n}μ_{i} and its standard deviation is s_{n}. If we normalize S_{n} by setting
then the distribution of Z_{n} converges towards the standard normal distribution N(0,1) as above.
Lindeberg condition
In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following weaker one: for every ε > 0
(where E( U : V > c) denotes the conditional expected value: the expected value of U given that V > c.) Then the distribution of the normalized sum Z_{n} converges towards the standard normal distribution N(0,1).
Non-independent case
There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.
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