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 will tend to be distributed according to a particular "attractor distribution". The most important and famous result is simply called The Central Limit Theorem which states that if the summed variables have a finite variance then they will be approximately normally distributed. Since many real processes yield distributions with finite variance, this explains the ubiquity of the normal distribution.

Lindeberg condition

In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following weaker one (from Lindeberg in 1920). For every ε > 0

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:

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lim_{n o infty} sum_{i = 1}^{n} mbox{E}left(

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rac{(X_i - mu_i)^2}{s_n^2}

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:

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left| X_i - mu_i ight| > epsilon s_n

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ight) = 0

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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 Zn converges towards the standard normal distribution N(0,1).

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