# central limit theorem

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## cen·tral lim·it the·o·rem

the sum (or average) of n realizations of the same process, provided only that it has a finite variance, will approach the gaussian distribution as n becomes indefinitely large. This theory provides a broad warrant for the use of normal theory even for nongaussian data. In the form stated here, it constitutes the classical version; more general versions allow serious relaxation of the usual assumptions.
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Theorem 21 then establishes a quantum central limit theorem for these observables.
The Central Limit Theorem accounts for the popularity of [bar.x] as a measure of central tendency.
We note that although the derivation here is based on the approach to the Gaussian distribution on the basis of the central limit theorem, the same type Edgeworth expansion can also be found in the context of nonlinear evolution of density fluctuations starting from random Gaussian linear fluctuations.
Fischer, A History of the Central Limit Theorem: From Classical to Modern Probability Theorem, Springer, New York, NY, USA, 1st edition, 2011.
Peng, "A new central limit theorem under sublinear expectations," In press, http://arxiv.org/abs/0803.2656.
Kerov s central limit theorem for the Plancherel measure on Young diagrams.
Describe in your own words the (celebrated) Central Limit Theorem. What assumptions must be met in order to use its conclusions?
The life and times of the central limit theorem, 2d ed.
A second argument sometimes mentioned in support of the log-normal distribution is based on the central limit theorem, which implies that under certain regularity conditions the distribution of a product of random variables will approach log normality as the number of terms increases.
The formulation of an accompanying central limit theorem takes into account that there is no convenient notion of subtraction for convex bodies, and so the identification with support functions is used:
Note that the sum in the denominator above converges to 1/2 by the Central Limit Theorem.
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