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The Splendors and Miseries of MartingalesPaul Lévy’s Perspective on Jean Ville and Martingales

The Splendors and Miseries of Martingales: Paul Lévy’s Perspective on Jean Ville and Martingales [As the first part of this chapter explains, Paul Lévy’s theory of martingales was about extending the law of large numbers and other theorems about sequences of independent random variables to dependent random variables, Lévy showed that this extension is possible when each random variable has mean zero given the preceding ones. Under this condition, the sequence of cumulative sums is a martingale as Jean Ville and later Joseph L. Doob used the word, but Lévy never focused on this sequence of cumulative sums as a mathematical object. In this respect, his was not a theory of martingales. Moreover, he never showed much interest in the properties of martingales studied by Ville and Doob. The second part of the chapter describes Lévy’s troubled relationship with Ville and his disdain for Ville’s mathematical work. We find insights into Lévy’s attitude towards Ville in the decades-long correspondence Lévy sustained with Maurice Fréchet.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

The Splendors and Miseries of MartingalesPaul Lévy’s Perspective on Jean Ville and Martingales

Part of the Trends in the History of Science Book Series
Editors: Mazliak, Laurent; Shafer, Glenn
Mazliak, Laurent
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
123 –145
DOI
10.1007/978-3-031-05988-9_6
Publisher site
See Chapter on Publisher Site

Abstract

[As the first part of this chapter explains, Paul Lévy’s theory of martingales was about extending the law of large numbers and other theorems about sequences of independent random variables to dependent random variables, Lévy showed that this extension is possible when each random variable has mean zero given the preceding ones. Under this condition, the sequence of cumulative sums is a martingale as Jean Ville and later Joseph L. Doob used the word, but Lévy never focused on this sequence of cumulative sums as a mathematical object. In this respect, his was not a theory of martingales. Moreover, he never showed much interest in the properties of martingales studied by Ville and Doob. The second part of the chapter describes Lévy’s troubled relationship with Ville and his disdain for Ville’s mathematical work. We find insights into Lévy’s attitude towards Ville in the decades-long correspondence Lévy sustained with Maurice Fréchet.]

Published: Oct 18, 2022

Keywords: History of probability theory; Martingales; Dependent random variables; Paul Lévy; Jean Ville; Primary: 01A60; 60-03; Secondary: 60G42; 60G44

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