• Bernoulli
  • Volume 19, Number 4 (2013), 1088-1121.

A Tricentenary history of the Law of Large Numbers

Eugene Seneta

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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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Bernoulli, Volume 19, Number 4 (2013), 1088-1121.

First available in Project Euclid: 27 August 2013

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


Seneta, Eugene. A Tricentenary history of the Law of Large Numbers. Bernoulli 19 (2013), no. 4, 1088--1121. doi:10.3150/12-BEJSP12.

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