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

ISBN-13: 9781461427018 / Angielski / Miękka / 2012 / 402 str.

Hans Fischer

Opis

This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.

Recenzje

From the book reviews:

"Fischer provides thorough mathematical descriptions of the development of the central limit theorem as it evolves with increasing mathematical rigor. ... Fischer has probably written what will be the definitive history of the central limit theorem for many years to come. ... Fischer overflows with detail, insight and excellent commentary." (David Bellhouse, Historia Mathematica, Vol. 39, 2012)

"The book will be of interest not only to professionals in the area of probability and statistics but to a wider audience. ... 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. ... the book would be an excellent source for student projects on topics from probability and its applications." (Jordan M. Stoyanov, Zentralblatt MATH, Vol. 1226, 2012)

"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. ... Hans Fischer ... 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, The Mathematical Association of America, February, 2011)

"The history of the CLT deserves a place of its own, and this book by Hans Fischer is the best ... in tracing its development in meticulous historical detail and with mathematical precision. ... The book by Hans Fischer is highly recommended as a well-researched comprehensive history of the CLT. One finds here the story of a galaxy of brilliant mathematicians ... engaging in the quest for, and debates on, the true meaning and the correct derivation of a beautiful intriguing result." (Rabi Bhattacharya, SIAM Review, Vol. 53 (4), 2011)

Spis treści

Preface.- Introduction.- The central limit theorem from laplace to cauchy: changes in stochastic objectives and in analytical methods.- The hypothesis of elementary errors.- Chebyshev's and markov's contributions.- The way towards modern probability.- General limit problems.- Conclusion: the central limit theorem as a link between classical and modern probability.- Index.- Bibliography

Blurb

This study aims to embed the history of the central limit theorem within the history of the development of probability theory from its classical to its modern shape, and, more generally, within the corresponding development of mathematics. The history of the central limit theorem is not only expressed in light of "technical" achievement, but is also tied to the intellectual scope of its advancement. The history starts with Laplace's 1810 approximation to distributions of linear combinations of large numbers of independent random variables and its modifications by Poisson, Dirichlet, and Cauchy, and it proceeds up to the discussion of limit theorems in metric spaces by Donsker and Mourier around 1950. This self-contained exposition additionally describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The importance of historical connections between the history of analysis and the history of probability theory is demonstrated in great detail. With a thorough discussion of mathematical concepts and ideas of proofs, the reader will be able to understand the mathematical details in light of contemporary development. Special terminology and notations of probability and statistics are used in a modest way and explained in historical context.