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The Life and Times of the Central Limit Theorem

William J. Adams
American Mathematical Society/London Mathematical Society
Publication Date: 
Number of Pages: 
History of Mathematics 35
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
John D. Cook
, on

The Life and Times of the Central Limit Theorem chronicles the history of the Central Limit Theorem (CLT) from its earliest beginnings to its mature form. The book’s author, William J. Adams, tells the story from the work of Abraham de Moivre in 1733 to the work of Aleksandr Lyapunov around 1900. Adams includes two expository papers, one by William Feller and another by Lucien le Cam, to cover the development of the CLT to its final form around 1935. As Feller points out, the “final” form of the theorem was not the end of research around the CLT. However, by 1936 necessary and sufficient conditions for the CLT were published and in a sense the theorem had reached its final form.

Roughly speaking, the CLT says that the sum of a large number of independent random variables X1 + X2 + ... + Xn has an approximately normal (Gaussian) distribution. This statement begs numerous questions. What are the requirements on the distributions Xi that go into the sum in order for the theorem to hold? Is independence necessary? Must the random variables be identically distributed, and if not, how different can they be? In what sense does the approximation converge? Could a sum approach some other limiting distribution other than the normal distribution? These are some of the questions that were resolved over the two centuries between the first hints of the CLT and its mature form.

The following presentation sequence is typical of a contemporary probability course.

  1. Introduce the normal distribution as the distribution with density
    (2π)–½ exp(–x2/2).
  2. Present the CLT, proving a special case of the theorem using moment generating functions.
  3. Remark that you can approximate binomial probabilities using a normal distribution.

The historical development was quite different than the sequence above, almost the reverse. The CLT began with the problem of computing binomial probabilities. de Moivre discovered that such probabilities could be approximated using integrals of the form exp(–x2) and proved a very special case of the CLT. Only later did anyone think of normalizing exp(–x2) to form the density of a probability distribution. And although de Moivre did have the idea of using a generating function for binomial probabilities, the technique of using moment generating functions would not appear until later. Also, the term “central limit theorem” did not arrive until Pólya so named the theorem in 1920.

Adams concludes what he calls the “early life and middle years” of the CLT with the work of Lyapunov and includes four papers of Lyapunov in an appendix. Lyapunov’s version of the CLT was much closer to its final form than to its embryonic form due to de Moive. Perhaps more important than the theorems he proved was the technique he developed, that of characteristic functions.

The latter half of Adams’ book, what he calls “the modern era”, consists of two expository papers: “The fundamental limit theorems in probability” by William Feller, and “The Central Limit Theorem around 1935” by Lucien le Cam.

Feller’s paper contains some historical detail but is primarily mathematical rather than historical. Also, his paper is not limited to the CLT but is also concerned with a related theorem, the law of the iterated logarithm. Le Cam’s paper quickly summarizes the entire history of the CLT but as the title implies the paper concentrates on the endgame, the work of William Feller, Paul Lévy, and Harald Cramér around 1935.

The Life and Times of the Central Limit Theorem is ostensibly the history of one theorem, but it touches on major themes in the development of probability, statistics, and modern analysis. And while it is ultimately a history book, it contains a generous portion of precise mathematics. Someone not interested in history would benefit from reading the book, especially Feller’s paper, in order to learn the nuances of various formations of the CLT.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.

Early life and middle years

  • A seed is sown
  • Approximation by integrals of $e^{-x^{2}}$
  • Impetus provided by the theory of errors of observation
  • Impetus provided by mathematical astronomy
  • The flowering of the central limit theorem begins
  • The development of the hypothesis of elementary errors
  • The emergence of an abstract central limit theorem
  • Chebyshev's pupils: A. A. Markov and A. M. Lyapunov
  • Bibliography

The modern era

  • W. Feller, The fundamental limit theorems in probability
  • L. Le Cam, The central limit theorem around 1935
  • H. F. Trotter, J. L. Doob, David Pollard, and L. Le Cam, Comments and rejoinder


  • A. M. Lyapunov, On a theorem in probability theory
  • A. M. Lyapunov, On a theorem in probability theory
  • A. M. Lyapunov, A general proposition in probability theory
  • A. M. Lyapunov, A new form of a theorem on the limit of a probability
  • Index