# What are Central Limit Theorems and why are they called so?

I know two opinions:

1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit distribution of fluctuations is not Gaussian, we should not call such statement CLT.

2) "Central" comes from "fluctuations around centre (=average)", and any theorem about limit distribution of such fluctuations is called CLT.

Which is correct?

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The way I see it is that the space of all probability distributions (satisfying the conditions required for CLT) has a distinguished point, namely the Gaussian. That's the center of the space. –  Deane Yang Oct 29 '10 at 13:54
I'm assuming of course that one has normalized the mean and variance. –  Deane Yang Oct 29 '10 at 13:55
Wikipedia (en.wikipedia.org/wiki/Central_limit_theorem#History) suggests #1: "The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper.[7](Le Cam 1986) Pólya referred to the theorem as "central" due to its importance in probability theory." –  Qiaochu Yuan Oct 29 '10 at 14:01
In my probability theory book (the one by H. Bauer) the name is also contributed to George Pólya, but nothing is written about the meaning. –  Someone Oct 29 '10 at 14:48
Both are correct. The first one if one uses "history" to evaluate correctness, and the second one if one uses "closeness to the center" to evaluate "correctness" ;-) –  Suvrit Oct 29 '10 at 18:33

One of my teacher in Probability once told us that this name (Central Limit Theorem) was just used (at the beginning) to stress the importance of the result -which plays a central role in the theory. Besides, the ambiguity led to several different translations, corresponding to both interpretations of the term "central". (e.g in French, we can find "théorème central limite" and "théorème de la limite centrale")

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Interesting. I had though that 2) was the reason, but I guess it was just speculation. By analogy with the usage "central tendency" for discussion of mean, median, mode, etc... –  Gerald Edgar Oct 29 '10 at 14:40
The German version clearly shows that "central" refers to the theorem, not to the limit. A correct French translation would be "théorème central de la limite", which I have never seen. –  Laurent Moret-Bailly Oct 29 '10 at 15:29
I was told recently by Firas Rassoul-Agha that 1) is the reason for the name. One of the French translations above (I'm not sure which since my French is bad) emphasizes that the theorem is the Central (i.e., fundamental) limit theorem in probability. Firas claimed that the other French translation arose by translating Central Limit Theorem back from English to French - thus obscuring the original meaning. –  Jon Peterson Oct 29 '10 at 15:59

From the introduction to History of the Central Limit Theorem: From Laplace to Donsker by Hans Fischer:

The term “central limit theorem” most likely traces back to Georg Pólya. As he recapitulated at the beginning of a paper published in 1920, it was “generally known that the appearance of the Gaussian probability density $e^{-x^2}$” in a great many situations “can be explained by one and the same limit theorem,” which plays “a central role in probability theory” [Pólya 1920, 171]. Laplace had discovered the essentials of this fundamental theorem in 1810, and with the designation “central limit theorem of probability theory,” which was even emphasized in the paper’s title, Pólya gave it the name that has been in general use ever since.

Fischer refers to the paper by G. Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift, 8 (1920), pp. 171–181.

Edit. The paper is reprinted in George Pólya: Collected Papers, Volume 4, MIT Press, 1984. R.M.Dudley mentions in his comment on the paper that

Although the name "central limit theorem" for the normal limit law seems to have been articulated in the mathematical folklore by 1920, Feller in his famous text attributes to Pólya the first written use of this term.

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There's a nice little book on this subject: The Life and Times of the Central Limit Theorem. The book goes over the history of the theorem from its embryonic form to its more or less final form 200 years later. It also gives precise statements of some of the modern variations of the theorem and indicates directions of research.

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