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The different forms and functions of the central limit theorem in the development from classical to modern probability theory. (Die verschiedenen Formen und Funktionen des zentralen Grenzwertsatzes in der Entwicklung von der klassischen zu der modernen Wahrscheinlichkeitsrechnung.) (German) Zbl 1015.60004

Aachen: Shaker Verlag. München: Univ. München, x, 307 S. (2000).
This study discusses the historical development of the probabilistic central limit theorem from about 1810 through 1940. The central limit theorem was originally deduced by Laplace as a statement about approximations for 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.
Main topics: Laplace’s way to the central limit theorem; Laplace’s method of approximations; applications of the central limit theorem by Laplace and by his successors; Poisson’s modifications; reconstruction of Dirichlet’s proof; the central limit theorem in Cauchy’s and Bienaymé’s controversy over the method of least squares; the development of the hypothesis of elementary errors from Hagen to Edgeworth, and its meaning for the “statistics of variations”; roots of the theory of moments; theory of moments and central limit theorem in Chebyshev’s and Markov’s works; first steps into modern probability by Lyapunov around 1900; contributions by Pólya, v. Mises, Lindeberg, Lévy, Bernshtein, and Cramér during the 1920s; generalizations towards dependent random variables and nonnormal limit laws; Lévy’s and Feller’s necessary conditions, discussion of Lévy’s original proof; general limit problems.

MSC:

60-03 History of probability theory
60F05 Central limit and other weak theorems
01A50 History of mathematics in the 18th century
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
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