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## Central limit theorem
The most important and famous result is simply called The reader may find it helpful to consider this illustration of the central limit theorem.
## "The" central limit theorem
Let X,_{2}X,... 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. _{3}
Consider the sum : 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
Z_{n} converges towards the standard normal distribution N(0,1)
as n approaches ∞. This means: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have
If the third central moment E((
*picture of a distribution being "smoothed out" by summation would be nice*
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 "paradox": by adding many independent identically distributed
More precisely: the fact that, in this case, for every n there is a z such that Pr( ## Alternative statements of the theoremThe 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. ## Lyapunov conditionAssume that the third central moments
n, and that
S=_{n}X. The expected value of _{1}+...+X_{n}S_{n} is m_{n} = ∑_{i=1..n}μ_{i} and its standard deviation is s_{n}. If we normalize S_{n} by setting
Z_{n} converges towards the standard normal distribution N(0,1) as above.## Lindeberg condition(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## External links
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