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The Life and Times of the Central Limit Theorem: Second Edition
William J. Adams, Pace University, New York, NY
A co-publication of the AMS and the London Mathematical Society.
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History of Mathematics
2009; 195 pp; hardcover
Volume: 35
ISBN-10: 0-8218-4899-2
ISBN-13: 978-0-8218-4899-9
List Price: US$56 Member Price: US$44.80
Order Code: HMATH/35

The study of any topic becomes more meaningful if one also studies the historical development that resulted in the final theorem. ... This is an excellent book on mathematics in the making.

--Philip Peak, The Mathematics Teacher, May, 1975

I find the book very interesting. It contains valuable information and useful references. It can be recommended not only to historians of science and mathematics but also to students of probability and statistics.

--Wei-Ching Chang, Historica Mathematica, August, 1976

In the months since I wrote ... I have read it from cover to cover at least once and perused it here and there a number of times. I still find it a very interesting and worthwhile contribution to the history of probability and statistics.

--Churchill Eisenhart, past president of the American Statistical Association, in a letter to the author, February 3, 1975

The name Central Limit Theorem covers a wide variety of results involving the determination of necessary and sufficient conditions under which sums of independent random variables, suitably standardized, have cumulative distribution functions close to the Gaussian distribution. As the name Central Limit Theorem suggests, it is a centerpiece of probability theory which also carries over to statistics.

Part One of The Life and Times of the Central Limit Theorem, Second Edition traces its fascinating history from seeds sown by Jacob Bernoulli to use of integrals of $$\exp (x^2)$$ as an approximation tool, the development of the theory of errors of observation, problems in mathematical astronomy, the emergence of the hypothesis of elementary errors, the fundamental work of Laplace, and the emergence of an abstract Central Limit Theorem through the work of Chebyshev, Markov and Lyapunov. This closes the classical period of the life of the Central Limit Theorem, 1713-1901.

The second part of the book includes papers by Feller and Le Cam, as well as comments by Doob, Trotter, and Pollard, describing the modern history of the Central Limit Theorem (1920-1937), in particular through contributions of Lindeberg, Cramér, Lévy, and Feller.

The Appendix to the book contains four fundamental papers by Lyapunov on the Central Limit Theorem, made available in English for the first time.

Undergraduate students, graduate students and research mathematicians interested in the history of mathematics, especially of probability theory.

Reviews

"[The second edition] is a refinement of its first edition...interesting to read."

-- Fuchang Gao, Mathematical Reviews

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
Appendix
• 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