Central limit theorems are a set of weakconvergence 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.

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 normalize S_{n} by setting
Then the distribution of 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((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 BerryEsséen theorem[?]).
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 "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 nonzero! How is it possible to get negative numbers from adding only positives? The key lies in the word "approximately". The sum of positive variables is of course always positive, but it is very well approximated by a normal variable (which indeed has a very tiny probability of being negative).
More precisely: the fact that, in this case, for every n there is a z such that Pr(Z_{n} ≤ z) = 0 does not contradict that for every z we have lim_{n→∞} Pr(Z_{n} ≤ z) > 0, because in the first case z may depend on n and in the second case n is increased for a fixed z.
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.
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
\lim_{n \to \infty} \sum_{i = 1}^{n} \mbox{E}\left( \frac{(X_i  \mu_i)^2}{s_n^2} : \left X_i  \mu_i \right > \epsilon s_n \right) = 0</math>
(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).
There are some theorems which treat the case of sums of nonindependent variables, for instance the mdependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.
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