# central limit theorem

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## central limit theorem

n
(Statistics) statistics the fundamental result that the sum (or mean) of independent identically distributed random variables with finite variance approaches a normally distributed random variable as their number increases, whence in particular if enough samples are repeatedly drawn from any population, the sum of the sample values can be thought of, approximately, as an outcome from a normally distributed random variable
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Even though we might not know the shape of the distributions where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal distribution.
Quantum central limit theorems are quantum analogs of the classical central limit theorem, which deal with observables from a quantum probability point of view.
Fournier [8] uses the quadratic Vaserstein distance for the approximation of the Euler scheme and the results of Rio [9] which gives a very precise rate of convergence for the central limit theorem in Vaserstein distance.
Bolthausen, "A central limit theorem for two-dimensional random walks in random sceneries," Annals of Probability, vol.
Keywords: Markov source, variance, covariance, independence, Hamming weight, Matrix-Tree Theorem, transducer, central limit theorem
An interactive apparatus developed by Hayrapetyan and Kuruvilla (2015) gives instructors and students a tool to visualize and "feel" the Central Limit Theorem. The tool allows the user to select a distribution (e.g., normal, uniform, skewed, or random), the sample size and see how the sampling distribution of the mean gradually becomes approximately normal.
Then, if [mu] = E([x.sub.i])and[[sigma].sup.2] = Var([X.sub.i]), the Central Limit Theorem states that:
The Edgeworth expansion (17) provides such a framework to incorporate them around the normal distribution, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in relation with the central limit theorem. Here and hereafter, the variable y (as well as [y.sub.i] or [y.sub.p] introduced later) denotes a normalized quantity with a vanishing mean and a unit variance such as y [equivalent to] x/[[sigma].sub.x] with [[sigma].sub.x] = [absolute value of <[x.sup.2]>).
The book wraps up with transformation of random variables, modes of convergence, the weak law of large numbers, and the central limit theorem, with a final chapter giving an overview of statistical inference.
Reference [28] shows that, if each layer data conforms to Lyapunov condition, the sum of each layer data will be in accordance with the application conditions of the central limit theorem; that is, the sum of each layer data will obey the normal distribution.
The Central Limit Theorem explains why this occurs and why the problem is unavoidable.
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