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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.
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}} .
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.
5.1 Proof using Chebyshev's inequality assuming finite variance. ... The limit e it μ is the ... Central limit theorem;
At the end of the 1960s, unsatisfied with the by-then known proofs of a specific central limit theorem, Charles Stein developed a new way of proving the theorem for his statistics lecture. [2] His seminal paper was presented in 1970 at the sixth Berkeley Symposium and published in the corresponding proceedings. [1]
Then according to the central limit theorem, the distribution of Z n approaches the normal N(0, 1 / 3 ) distribution. This convergence is shown in the picture: as n grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.
The central limit theorem can provide more detailed information about the behavior of than the law of large numbers. For example, we can approximately find a tail probability of M N {\displaystyle M_{N}} – the probability that M N {\displaystyle M_{N}} is greater than some value x {\displaystyle x} – for a fixed value of N {\displaystyle N} .
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]