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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}} .
By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal.
Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.
Cameron–Martin theorem; Campbell's theorem (probability) Central limit theorem; Characterization of probability distributions; Chung–ErdÅ‘s inequality; Condorcet's jury theorem; Continuous mapping theorem; Contraction principle (large deviations theory) Coupon collector's problem; Cox's theorem; Cramér–Wold theorem; Cramér's theorem ...
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
The proof is based on the distribution of parity vectors and uses the central limit theorem. In 2019, Terence Tao improved this result by showing, using logarithmic density , that almost all (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to ...
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]