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.

Lyapunov condition

See also Lyapunov's central limit theorem.

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Let Xn be a sequence of independent random variables defined on the same probability space. Assume that Xn has finite expected value μn and finite standard deviation σn. We define

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:s_n^2 = sum_{i = 1}^n sigma_i^2.

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Assume that the third central moments

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:r_n^3 = mbox{E}left({left| X_n - mu_n ight|}^3 ight)

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are finite for every n, and that

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:lim_{n o infty} rac{r_n}{s_n} = 0.

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(This is the Lyapunov condition).

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We again consider the sum Sn=X1+...+Xn. The expected value of Sn is mn = ∑i=1..nμi and its standard deviation is sn. If we normalize Sn by setting

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:Z_n = rac{S_n - m_n}{s_n}

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then the distribution of Zn converges towards the standard normal distribution N(0,1) as above.

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