## Monday, 10 March 2014

### Lecture 23

On the way home today, I realised that I had forgotten to tell you why the Central limit theorem has that name. See below. First, here is a little more history about this important theorem.

Henk Tijms writes in his book, Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press, 2004,

"The Central Limit Theorem for Bernoulli trials was first proved by Abraham de Moivre and appeared in his book, The Doctrine of Chances, first published in 1718. He 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.

De Moivre spend his years from age 18 to 21 in prison in France because of his Protestant background. When he was released he left France for England, where he worked as a tutor to the sons of noblemen. Newton had presented a copy of his Principia Mathematica to the Earl of Devonshire. The story goes that, while de Moivre was tutoring at the Earl's house, he came upon Newton's work and found that it was beyond him. It is said that he then bought a copy of his own and tore it into separate pages, learning it page by page as he walked around London to his tutoring jobs. De Moivre frequented the coffeehouses in London, where he started his probability work by calculating odds for gamblers. He also met Newton at such a coffeehouse and they became fast friends. De Moivre dedicated his book to Newton."

I was looking at the Wikipedia article on the Central limit theorem. It mentions two things that I did not already know, but which would be suitable for the television programme QI.

1. There is a quite interesting explanation of why the term "Central" is used.

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper. 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".

Personally, I had always thought it was the second of these, but the first is also plausible.

2. There is a quite interesting Cambridge connection.

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. Consequently, Turing's dissertation was never published.