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.

Related Topics:
Probability theory - Independent - Random variable - Normally distributed - Variance - Distribution

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Several generalizations for finite variance 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. Also, a generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1/|x|α+1 with 0 

Related Topics:
Lindeberg - Lyapunov - Gnedenko - Kolmogorov - Lévy distribution

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The reader may find it helpful to consider this illustration of the central limit theorem.

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