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A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved.
Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem. The main technique involved in making calculations with a characteristic function is recognizing the function as the characteristic function of a particular distribution.
This theorem can be used to disprove the central limit theorem holds for by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for X k {\displaystyle X_{k}} .
The method of moments was introduced by Pafnuty Chebyshev in 1887 in the proof of the central limit theorem. The idea of matching empirical moments of a distribution to the population moments dates back at least to Karl Pearson.
Comparison of probability density functions, () for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black ...
In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions.
The method of moments was introduced by Pafnuty Chebyshev for proving the central limit theorem; Chebyshev cited earlier contributions by Irénée-Jules Bienaymé. [2] More recently, it has been applied by Eugene Wigner to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrices. [3]
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. [ 14 ] The first published complete proof (in French) of the GCLT was in 1937 by Paul Lévy . [ 15 ]