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Central limit theorem

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The statement of the theorem

The central limit theorem (CLT) explains the ubiquitous occurrence of the normal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics." The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let X1,X2,… X_{1},X_{2},\dots \, be independent random variables with mean μ\mu and variance σ2>0. \sigma ^{2}>0.\, Then the sequence of random variables Zn=∑i=1n(Xi−μ)σn Z_{n}={\frac {\sum _{i=1}^{n}(X_{i}-\mu )}{\sigma {\sqrt {n}}}}\, converges in distribution to a standard normal random variable. For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem. For example, the distributions with finite first, second, and third moment from the exponential family; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly or may not work at all: in such cases one may use the Generalized Central Limit Theorem (GCLT). - ^David Williams, "Probability with martingales", Cambridge 1991/2008
Source: Wikipedia