# Central Limit Theorem

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## Central Limit Theorem

The Law of Large Numbers states that as a sample of independent, identically distributed random numbers approaches infinity, its probability density function approaches the normal distribution. See: Normal Distribution.

## Central Limit Theorem

In statistics, a theory stating that as the sample size of identically distributed, random numbers approaches infinity, it is more likely that the distribution of the numbers will approximate normal distribution. That is, the mean of all samples within that universe of numbers will be roughly the mean of the whole sample.
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Sweeting, "Uniform asymptotic normality of the maximum likelihood estimator," Annals of Statistics, vol.
This approximation is reasonable, since [[??].sup.measure.sub.A] is already strongly consistent by assuming only the ergodicity of [[PHI].sub.1] (which is a milder condition than the ones required for the asymptotic normality of [[??].sub.A,i]).
As in Section 3.1, sufficient conditions for asymptotic normality are: i) [[delta].sup.2.sub.i] > 0, and ii) the SDFs of the different models are distinct.
The method of moments applies to all these cases and establishes the central limit theorems; similar details are given as in  (the asymptotic normality of the number of leaves being already proved there as a special case).
Mahmoud, Smythe and Szymanski (12) used a representation with a generalized Polya urn to prove the asymptotic normality (1.5).
(1996) Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH (1,1).
The consistency and asymptotic normality of the QMLE has been established only for specific special cases of the ARFIMA and/or FIGARCH model.
These confidence intervals are constructed based on the asymptotic normality of the estimators for the sub-indices of the [C.sub.pk] index, and the process distribution need not be normal nor be known.
This assumption can be harder to justify than the asymptotic normality demanded by the t test, and is rarely evaluated (Petranka 1990).
The topics include algebraic methods, discrete geometric methods, analytic methods, asymptotic normality in enumeration, trees, planar maps, graphs, words, tilings, lattice path enumeration, permutation classes, parking functions, standard Young tableaux, and computer algebra.
From Lyapunov's theorem, the asymptotic normality of the penalized spline estimator [??](x) with [[??].sub.opt] can be derived under the same assumption as Theorem 2 and some additional mild conditions.

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