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The Splendors and Miseries of MartingalesThe Dawn of Martingale Convergence: Jessen’s Theorem and Lévy’s Lemma

The Splendors and Miseries of Martingales: The Dawn of Martingale Convergence: Jessen’s Theorem... [Jessen’s theorem and Lévy’s lemma, which both date from 1934, are the earliest known general formulations of the martingale convergence theorem. Børge Jessen worked within Lebesgue’s theory of integration; he saw his theorem as an extension of the Fubini-Lebesgue theorem of 1907–1920. Paul Lévy’s vision was probabilistic; he saw his lemma as an extension of Borel’s strong law of large numbers of 1909. This chapter reviews how the two arrived at their results. Jessen published his theorem in 1934, and it helped inspired Lévy’s formulation of his lemma. In letters between the two authors, each wanted to see the other’s result as a trivial consequence of their own. Jessen sought a level of abstraction that proved unattainable, but his interaction with Lévy can be seen as the origin of a now standard version of the martingale convergence theorem. This standard version was first stated by Erik Sparre AndersenAndersen, Erik Sparre and Jessen in 1946, and its standard proof relies on ideas that Jessen and Lévy developed in their correspondence. The correspondence is reproduced in the present volume in the chapter entitled “Analysis or Probability? Eight letters between Børge JessenJessen, Børge and Paul LévyLévy, Paul”.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

The Splendors and Miseries of MartingalesThe Dawn of Martingale Convergence: Jessen’s Theorem and Lévy’s Lemma

Part of the Trends in the History of Science Book Series
Editors: Mazliak, Laurent; Shafer, Glenn
Eid, Salah
The Splendors and Miseries of Martingales — Oct 18, 2022
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Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
ISBN
978-3-031-05987-2
Pages
67 –104
DOI
10.1007/978-3-031-05988-9_4
Publisher site
See Chapter on Publisher Site

Abstract

[Jessen’s theorem and Lévy’s lemma, which both date from 1934, are the earliest known general formulations of the martingale convergence theorem. Børge Jessen worked within Lebesgue’s theory of integration; he saw his theorem as an extension of the Fubini-Lebesgue theorem of 1907–1920. Paul Lévy’s vision was probabilistic; he saw his lemma as an extension of Borel’s strong law of large numbers of 1909. This chapter reviews how the two arrived at their results. Jessen published his theorem in 1934, and it helped inspired Lévy’s formulation of his lemma. In letters between the two authors, each wanted to see the other’s result as a trivial consequence of their own. Jessen sought a level of abstraction that proved unattainable, but his interaction with Lévy can be seen as the origin of a now standard version of the martingale convergence theorem. This standard version was first stated by Erik Sparre AndersenAndersen, Erik Sparre and Jessen in 1946, and its standard proof relies on ideas that Jessen and Lévy developed in their correspondence. The correspondence is reproduced in the present volume in the chapter entitled “Analysis or Probability? Eight letters between Børge JessenJessen, Børge and Paul LévyLévy, Paul”.]

Published: Oct 18, 2022

Keywords: History of probability; Jessen’s theorem; Lévy’s lemma; Martingale convergence theorem; Transfer principle

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