Editing Central limit theorem (section)

==History==
Dutch mathematician [[Henk Tijms]] writes:<ref name=Tijms/>

{{blockquote|The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician [[Abraham de Moivre]] who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician [[Pierre-Simon Laplace]] rescued it from obscurity in his monumental work ''Théorie analytique des probabilités'', which was published in 1812.  Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician [[Aleksandr Lyapunov]] defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.}}

Sir [[Francis Galton]] described the Central Limit Theorem in this way:<ref>{{cite book|last=Galton|first= F. |date=1889 |title=Natural Inheritance |url=http://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm |page= 66}}</ref>

{{blockquote|I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.}}

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by [[George Pólya]] in 1920 in the title of a paper.<ref name=Polya1920>{{Cite journal|last=Pólya|first=George|author-link=George Pólya|year=1920|title=Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem|trans-title=On the central limit theorem of probability calculation and the problem of moments |journal=[[Mathematische Zeitschrift]]|volume=8|pages=171–181 |language=de |url=http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0008|doi=10.1007/BF01206525 |issue=3–4|s2cid=123063388}}</ref><ref name=LC1986/> Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word ''central'' in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".<ref name=LC1986/> The abstract of the paper ''On the central limit theorem of calculus of probability and the problem of moments'' by Pólya<ref name=Polya1920/> in 1920 translates as follows.

{{blockquote|text=The occurrence of the Gaussian probability density {{math|1 {{=}} ''e''<sup>−''x''<sup>2</sup></sup>}} in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by [[Aleksandr Lyapunov|Liapounoff]]. ... }}

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as [[Augustin-Louis Cauchy|Cauchy]]'s, [[Friedrich Bessel|Bessel]]'s and [[Siméon Denis Poisson|Poisson]]'s contributions, is provided by Hald.<ref name=Hald/> Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by [[Richard von Mises|von Mises]], [[George Pólya|Pólya]], [[Jarl Waldemar Lindeberg|Lindeberg]], [[Paul Lévy (mathematician)|Lévy]], and [[Harald Cramér|Cramér]] during the 1920s, are given by Hans Fischer.{{sfnp|Fischer|2011|loc=Chapter 2; Chapter 5.2}} Le Cam describes a period  around 1935.<ref name=LC1986/> Bernstein<ref name=Bernstein/> presents a historical discussion focusing on the work of [[Pafnuty Chebyshev]] and his students [[Andrey Markov]] and [[Aleksandr Lyapunov]] that led to the first proofs of the CLT in a general setting.

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, Cambridge|King's College]] at the [[University of Cambridge]].  Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.<ref>{{cite journal |first=S. L. |last=Zabell |title=Alan Turing and the Central Limit Theorem |journal=American Mathematical Monthly |volume=102 |year=1995 |issue=6 |pages=483–494 |doi=10.1080/00029890.1995.12004608 }}</ref>