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The Splendors and Miseries of Martingales pp 123–145Cite as

Birkhäuser

Paul Lévy’s Perspective on Jean Ville and Martingales

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Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

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.

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Notes

  1. 1.

    For more detailed treatments of this topic, see Lévy’s comments in his autobiography [2, 3, 34, 35, 39].

  2. 2.

    See for instance [43].

  3. 3.

    See [20, pp. 253ff] for details about Bernstein’s extensions of the central limit theorem to sums of dependent variables.

  4. 4.

    Bernstein’s interest in secular perturbations was inspired by a paper published by Bohl in 1909.

  5. 5.

    That is to say, Bernstein’s own Theorem 4 in [4].

  6. 6.

    On Hostinský’s beginnings in probability, see in particular [23].

  7. 7.

    The tortuous story of Fréchet and Lévy’s elections to the Academy can be followed in detail in [2].

  8. 8.

    We do not know when Khinchin had an occasion to discuss the matter with Lévy. The appearance of Khinchin’s name is interesting because beginning in the 1920s he had been one of the first readers and critics of von Mises’ collectives, which, despite some regrettable idealistic tendencies, were considered the approach to probability most compatible with the young USSR’s dialectical materialism. See [45] for a translation with commentary of the 1929 text by Khinchin on the subject. As late as 1952, in the icy final period of the Stalinist era, Khinchin again came back to this question in the rather controversial and ideological book Philosophical questions of contemporary physics with a chapter entitled The method of arbitrary functions and the battle against idealism in probability theory [19].

  9. 9.

    This book is part of the great Borelian project of the interwar period, the Treaty of probability and its application, which Borel launched at the beginning of the 1920s and published in successive volumes until 1939. In [15], the authors study the origins and the development of the Borelian project, and how Borel convened much of his network of past students of the École Normale to publish his lecture notes. About Ville, see in particular Sects. 2.2.5 and 3.1.5 in [15].

  10. 10.

    This means for their graduation.

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  1. Statistiques et Modélisation, Sorbonne Université. Laboratoire de Probabilités, Paris, France

    Laurent Mazliak

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  1. LPSM, Sorbonne Université, Paris, France

    Laurent Mazliak

  2. Rutgers Business School, Newark, NJ, USA

    Glenn Shafer

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Mazliak, L. (2022). Paul Lévy’s Perspective on Jean Ville and Martingales. In: Mazliak, L., Shafer, G. (eds) The Splendors and Miseries of Martingales. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-05988-9_6

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