Central Limit Theorem

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Then the distribution of ''Z''_''n'' Converges towards the Standard Normal Distribution N(0,1)
as ''n'' approaches ∞ (this is of N(0,1), then for every Real Number ''z'', we have

:$\backslash lim\_\{n\; o\; \backslash infty\}\; \backslash mbox\{Pr\}(Z\_n\; \backslash le\; z)\; =\; \backslash Phi(z),$
or, equivalently,

:
where
:$\backslash overline\{X\}\_n=S\_n/n=(X\_1+\backslash cdots+X\_n)/n$
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 ,

:

where ''o'' (''t²'' ) is " Little O Notation " for some function of ''t''  that goes to zero more rapidly than ''t²''. 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 X₁, X₂, ..., X_''n'' is just

:

By simple properties of characteristic functions, the characteristic function of ''Z''_''n'' is

:
\over 2n} + o\left({t^2 \over n} ight) ight]^n \, ightarrow \, e^{-t^2/2}, \quad n ightarrow \infty.

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 Convolution s):

(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 standardize ''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.

The Central Limit Theorem, as an approximation for a finite number of observations, 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 Central Limit theorem also applies to sums of independent and identical s is still a Discrete Random Variable , so that we are confronted to a Series of Discrete Random Variable s whose probability distribution converges towards a Probability Density Function corresponding to a continuous variable (namely the Normal Distribution ). This means that if we build an Organigram 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 organigram converges toward a gaussian curve as ''n'' approaches $\backslash infty$. 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.


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 Transform s,
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 independent 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


See also Lyapunov's Central Limit Theorem .

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

:$s\_n^2\; =\; \backslash sum\_\{i\; =\; 1\}^n\; \backslash sigma\_i^2.$

Assume that the third central moments