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A History of the Central Limit Theorem

Part of the Sources and Studies in the History of Mathematics and Physical Sciences Book Series

DOI: 10.1007/978-0-387-87857-7

ISBN: 978-0-387-87856-0

Publisher: Springer New York - Springer Journals

Aug 21, 2010

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Introduction

Fischer, Hans

Pages: 1 –16

[ The term “central limit theorem” most likely traces back to Georg PÓlya. As he recapitulated at the beginning of a paper published in 1920, it was “generally known that the appearance of the Gaussian probability density1\documentclass[12pt]{minimal} \usepackage{amsmath}...
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The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods

Fischer, Hans

Pages: 17 –74

[In 1812, Pierre-Simon de Laplace (1749–1827) published the first edition of his Théorie analytique des probabilités (henceforth simply abbreviated by TAP).1 With its typical problems, stochastic models, and analytic methods this book would considerably influence probability theory and...
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The Hypothesis of Elementary Errors

Fischer, Hans

Pages: 75 –138

[In the framework of classical probability theory, the primary objective was to calculate probabilities of certain events, with the aim of making“rational” decisions based on these probabilities. Error or frequency functions1 only played the role of auxiliary subjects. This paradigm, however,...
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Chebyshev’s and Markov’s Contributions

Fischer, Hans

Pages: 139 –189

[From an historical point of view, the treatment of the CLT by themethod of moments is strongly connected with the contributions of Chebyshev and Markov. According to the standard historical interpretation, both mathematicians with their contributions began to apply that mathematical rigor to...
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The Way Toward Modern Probability

Fischer, Hans

Pages: 191 –269

[Richard von Mises [1919a, 1] wrote in his pioneering paper “Fundamentalsätze der Wahrscheinlichkeitsrechnung”: The analytical theorems of probability theory are lacking—except for a few works by Russian mathematicians—the precision of formulation and reasoning which has long been a matter of...
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Lévy and Feller on Normal Limit Distributions around 1935

Fischer, Hans

Pages: 271 –314

[Lévy [1924, 17] had already stated that Lindeberg‘s condition for the CLT accorded particularly well with “la nature des choses.” Was he trying to say that, in a certain way, this conditionwas also necessary for convergence to the normal distribution? In fact, more than 10 years would pass...
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Generalizations

Fischer, Hans

Pages: 315 –352

[Lévy‘s and Feller‘s theorems of 1935 served as a paradigm for further work on sums of independent one- or multidimensional random variables, on the one hand. This strand of development largely preserved the “traditional” analytic orientation. On the other hand, generalizations toward martingales...
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Conclusion: The Central Limit Theorem as a Link Between Classical and Modern Probability Theory

Fischer, Hans

Pages: 353 –362

[During the period between the two world wars, modern probability theory emerged as a mathematical subdiscipline—with the typical formation of concepts, main theorems and methods—by integrating the subfields of axiomatics (including aspects of measure theory), strong laws of large numbers,...

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