A History of the Central Limit Theorem: From Classical to Modern Probability Theory (Sources and Studies in the History of Mathematics and Physical Sciences)

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. Introduction -- The Cental 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 Toward Modern Probability -- Lévy And Feller On Normal Limit Distributions Around 1935 -- Generalizations -- Conclusion: The Central Limit Theorem As A Link Between Classical And Modern Probability Theory. Hans Fischer. Includes Bibliographical References (p. 363-391) And Indexes. 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. Front Matter....Pages i-xvi Introduction....Pages 1-16 The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods....Pages 17-74 The Hypothesis of Elementary Errors....Pages 75-138 Chebyshev’s and Markov’s Contributions....Pages 139-189 The Way Toward Modern Probability....Pages 191-269 Lévy and Feller on Normal Limit Distributions around 1935....Pages 271-314 Generalizations....Pages 315-352 Conclusion: The Central Limit Theorem as a Link Between Classical and Modern Probability Theory....Pages 353-362 Back Matter....Pages 363-402