**History of Mathematics**

Volume: 35;
2009;
195 pp;

MSC: Primary 60;
Secondary 01

**Electronic ISBN: 978-1-4704-1806-9
Product Code: HMATH/35.E**

List Price: $56.00

AMS Member Price: $44.80

MAA Member Price: $50.40

#### Supplemental Materials

# The Life and Times of the Central Limit Theorem: Second Edition

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*William J. Adams*

A co-publication of the AMS and the London Mathematical Society

About the First Edition:

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.

#### Reviews & Endorsements

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

-- Fuchang Gao, Mathematical Reviews

#### Table of Contents

# Table of Contents

## The Life and Times of the Central Limit Theorem: Second Edition

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface to the second edition ix10
- Preface to the first edition xi12
- Part I. Early life and middle years 114
- A seed is sown 316
- Approximation by integrals of 𝑒^{-𝑥²} 1124
- Impetus provided by the theory of errors of observation 2134
- Impetus provided by mathematical astronomy 3144
- The flowering of the central limit theorem begins 3346
- The development of the hypothesis of elementary errors 3952
- The emergence of an abstract central limit theorem 4558
- Chebyshev’s pupils: A. A. Markov and A. M. Lyapunov 5366
- Bibliography 6780
- Part II. The modern era 7992
- W. Feller, The fundamental limit theorems in probability 8194
- L. Le Cam, The central limit theorem around 1935 115128
- H. F. Trotter, J. L. Doob, David Pollard, and L. Le Cam, Comments and rejoinder 139152
- Part III. Appendix 147160
- A. M. Lyapunov, On a theorem in probability theory 149162
- A. M. Lyapunov, On a theorem in probability theory 151164
- A. M. Lyapunov, A general proposition in probability theory 173186
- A. M. Lyapunov, A new form of a theorem on the limit of a probability 175188
- Index 193206
- Back Cover Back Cover1218