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September 2013 A Tricentenary history of the Law of Large Numbers
Eugene Seneta
Bernoulli 19(4): 1088-1121 (September 2013). DOI: 10.3150/12-BEJSP12


The Weak Law of Large Numbers is traced chronologically from its inception as Jacob Bernoulli’s Theorem in 1713, through De Moivre’s Theorem, to ultimate forms due to Uspensky and Khinchin in the 1930s, and beyond. Both aspects of Jacob Bernoulli’s Theorem: 1. As limit theorem (sample size $n\to\infty$), and: 2. Determining sufficiently large sample size for specified precision, for known and also unknown $p$ (the inversion problem), are studied, in frequentist and Bayesian settings. The Bienaymé–Chebyshev Inequality is shown to be a meeting point of the French and Russian directions in the history. Particular emphasis is given to less well-known aspects especially of the Russian direction, with the work of Chebyshev, Markov (the organizer of Bicentennial celebrations), and S.N. Bernstein as focal points.


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Eugene Seneta. "A Tricentenary history of the Law of Large Numbers." Bernoulli 19 (4) 1088 - 1121, September 2013.


Published: September 2013
First available in Project Euclid: 27 August 2013

zbMATH: 1277.60005
MathSciNet: MR3102545
Digital Object Identifier: 10.3150/12-BEJSP12

Keywords: Bienaymé–Chebyshev Inequality , J.V. Uspensky and S.N. Bernstein , Jacob Bernoulli’s Theorem , Markov’s Theorem , P.A. Nekrasov and A.A. Markov , Stirling’s approximation

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

Vol.19 • No. 4 • September 2013
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