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A History of the Central Limit Theorem: From Classical to Modern Probability Theory

Hans Fischer
Publication Date: 
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Sources and Studies in the History of Mathematics and Physical Sciences
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tom Schulte
, on

The history of mathematics is a favorite topic of mine and there are many fine books available that cover the topic broadly. It was a delight to savor this detailed study of “the” Central Limit Theorem alone as it arose and transformed in history. The scare quotes on the definite article are appropriate, as “central limit theorem” applies to a herd of statements on the convergence of probability distribution functions with increasing numbers of random variables. Author Hans Fischer ably guides the reader through the migrations of the CLT in its varied forms.

This work details the history of the central limit theorem and related probabilistic limit theorems roughly from 1810 through 1950, but focuses on 1810 to 1935. The story begins with the initial insights of Laplace, built upon by Poisson and Cauchy. The first two chapters of this earliest part of the telling are available on the author’s web site. The growth in definition and application of the CLT from this beginning comes from contributions of Chebyshev, Markov, von Mises, Pólya, Lindeberg, Lévy, Cramer, and others covered here. The theorems of Lévy and Feller in 1935 become a denouement, of sorts, to a historical tale not lacking in dramatic development. Along the way, Fischer examines questions of priority, motivation, rigor, and impact.

In teasing out the threads of an evolving CLT and the tools used to prove it, such as characteristic functions and moments, the author shines light on such broader topics as the shift from classical to mathematical probability during the Cauchy and Bienyamé era. Fischer goes into such detail as the unpublished lecture notes and drafts of Dirichlet to explore his specific approach to building on the work of Laplace.

Hans Fischer co-supervises the teaching of mathematics at Germany’s Catholic University Eichstätt-Ingolstadt and authors many papers on the history of mathematics. His skill in both these areas allows him to reveal here the historical development of this important theorem in a way that can easy be adapted to the lecture hall or used in independent study.

Tom Schulte helps actualize the powers of the Central Limit Theorem through designing Statistical Process Control (SPC) software for manufacturers at Plex Systems in Michigan.