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A history of the central limit theorem. From classical to modern probability theory. (English) Zbl 1226.60004
Sources and Studies in the History of Mathematics and Physical Sciences. New York, NY: Springer (ISBN 978-0-387-87856-0/hbk; 978-1-4614-2701-8/pbk; 978-0-387-87857-7/ebook). xvi, 402 p. EUR 109.95/net; £ 99.00; SFR 171.00 (2011).

For good reasons the central limit theorem (CLT) was, is and will be at the heart of probability theory, statistics and their applications. This rich, carefully written and well documented book tells us an amazing story, covering the period from around 1810 to 1950, i.e., from the work of Laplace on the CLT for independent Bernoulli random variables to the studies of Donsker and Mourier on functional CLTs.

Even if one thinks that one knows well, or up to some extend, how the CLT was developed, one can be sure, to find in this book a lot of new and interesting details. The “journey” starts in the time of de Moivre and Laplace and continues in the period of Poisson, Cauchy, Dirichlet and Gauss. It is more than interesting to learn about the new ideas and techniques used by these scientists and their followers. Their work has been motivated by the ambition to extend previously known results and answer new questions coming from applied areas. The next remarkable progress in the development of the CLT is due to works of Chebyshev, Markov, Lyapunov, Poincaré and Bernstein. It was a slow process, but the CLT was getting a general and rigorous form such that mathematicians of that time started to accept probability theory as mathematics. Further significant progress was made by von Mises, Lindeberg, Pólya, Lévy, Khintchine, Cramér, Gnedenko and Feller; and it was Kolmogorov who not only obtained first class limit theorems, but also based on achievements in measure theory and integration, suggested a very general and unified model of probability theory, a model which was and is universally accepted since 1933. The last stage of the “journey” in this book are studies around 1950 when the CLT was extended to cover sequences of random variables which are not independent are and not identically distributed. Convergence of stochastic processes based on such random sequences was investigated and functional CLTs were established.

At the end of the book, there is a very comprehensive reference list and index.

The book will be of interest not only to professionals in the area of probability and statistics but to a wider audience. The reader needs, of course, some knowledge in probability and calculus, occasionally in combinatorics, in order to follow the presentation. Everything is explained in historical context, so we can see that the progress in the CLT was going along and with progress in calculus and analysis. The author has been using a huge amount of sources and archives, including his own works, and he is successful in his goal to describe a comprehensive picture of the development of the CLT.

At any good university, the courses in probability include the CLT as one of the main topics. Such a course, at any level, can be made more interesting and attractive if the teacher uses even small parts of this book. Finally, the book would be an excellent source for student projects on topics from probability and its applications.

60-03Historical (probability theory)
01A55Mathematics in the 19th century
01A60Mathematics in the 20th century
60F05Central limit and other weak theorems