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Martingales in the Study of Randomness

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The Splendors and Miseries of Martingales

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

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Abstract

Martingales played an important role in the study of randomness in the twentieth century. In the 1930s, Jean Ville used martingales to improve Richard von Mises’s and Abraham Wald’s concept of an infinite random sequence, or collective. After the development of algorithmic randomness by Andrei Kolmogorov, Ray Solomonoff, Gregory Chaitin, and Per Martin-Löf in the 1960s, Claus-Peter Schnorr developed Ville’s concept in this new context. Along with Schnorr, Leonid Levin was a key figure in the development in the 1970s. While Schnorr worked with algorithmic martingales and supermartingales, Levin worked with the closely related concept of a semimeasure. In order to characterize the randomness of an infinite sequence in terms of the complexity of its prefixes, they introduced new ways of measuring complexity: monotone complexity (Schnorr and Levin) and prefix complexity (Levin and Chaitin).

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Notes

  1. 1.

    Ein Kollektivgegenstand ist eine Vielheit von gleichartigen Dingen, die nach einem veränderlichen Merkmal statistisch geordnet werden kann. [8, p. 96], in italics in the original.

  2. 2.

    In the original: Beispiele solcher unendlich gedachter Folgen sind die Ziehungen aus einer Urne, die etwa beim gewöhnlichen Lotto zu einem 5-dimensionalen Merkmalraum führen (Koordinaten sind die 5 Nummern einer Ziehung), oder die Moleküle eines Gases mit dem 3-dimensionalen Merkmalraum ihrer Geschwindigkeiten...[54, p. 70]

  3. 3.

    See for example §2 of Marie-France Bru and Bernard Bru’s chapter, “Borel’s denumerable martingales, 1909–1949”, in the present volume.

  4. 4.

    The letters between von Mises and Hausdorff on the topic are reproduced in [31, pp. 825–829]. Hausdorff also had difficulties with von Mises first axiom, but he thought the second axiom raised the greatest logical difficulties (p. 826). Jean Ville recalled that Maurice Fréchet also had the most difficulty with the second axiom; see the chapter of the present volume devoted to Pierre Crépel’s interview and correspondence with Ville. Reinhard Siegmund-Schultze has reviewed Pólya and Hausdorff’s objections and von Mises’s extensive interaction with Pólya [81]. Von Mises received a very negative reception at Göttingen in 1931 [81, p. 493]. We shortly discuss his reception at Geneva in 1937. His 1940 debate with the United States mathematician Joseph Doob, at a meeting of the Institute of Mathematical Statistics at Dartmouth, New Hampshire, is also notable [59, 61].

  5. 5.

    Popper’s exposition was not as mathematically precise as that of the other two authors, and the assertion that his proposal was exactly identical with theirs may be a simplification.

  6. 6.

    In the original: Die Festsetzung daß in einem Kollektiv jede Stellenauswahl die Grenzhäufigkeit unverändert läßt, besagt nichts anders als dies: Wir verabreden, daß, wenn in einer konkreten Aufgabe ein Kollektiv einer bestimmten Stellenauswahl unterworfen wird, wir annehmen wollen, diese Stellenauswahl ändere nichts an den Grenzwerten der relativen Häufigkeiten. Nichts darüber hinaus enthält mein Regellosigkeitsaxiom.

  7. 7.

    In a footnote to the Comptes rendus note, Wald says that he will give proofs in the seventh volume of Menger’s Ergebnisse, the volume for 1934–35. In a letter to von Mises dated 27 April 1936 (Papers of Richard von Mises, HUG 4574.5 Box 3, Folder 1936, Harvard University Archives), Wald states that the article with the proofs will appear in July 1936. But the notice of Wald’s talk on 13 February 1935, in the seventh volume of the Ergebnisse, states correctly that it is in the eighth volume, which did not appear in print until 1937 [88].

  8. 8.

    This comment appears in an endnote on p. 274 of the second German edition of Wahrscheinlichkeit, Statistik und Wahrheit [56]: “In der gleichen Richtung noch wesentlich weiter gehende Resultate bei A. Wald,...”. Wald is not otherwise mentioned in the book.

  9. 9.

    There is a subtle shift in terminology here. Von Mises had called a subsequence selection rule a Stellenauswahl, or “place selection". Wald calls it instead an Auswahlvorschrift, or “selection procedure” (in the note in French he used procédé de choix). The word Vorschrift, which can also be translated as “instruction”, puts an emphasis on the computational aspect of the selection that was missing from von Mises’s terminology.

  10. 10.

    For a history of Cournot’s principle and examples of statements embracing it by Jacques Hadamard, Paul Lévy, Maurice Fréchet, and Émile Borel, see [74]. The principle was named after Cournot by Fréchet around 1950.

  11. 11.

    For centuries the word martingale has referred to the strategy for betting that doubles one’s bet after every loss, and it was also used for more complicated strategies. See Roger Mansuy’s chapter on the word martingale and Glenn Shafer’s chapter “Martingales at the casino” in the present volume.

  12. 12.

    An asymptotic property of \(\omega _1\omega _2\ldots \) is one that does not depend on any finite prefix. In 1933 [32], Kolmogorov had shown that the probability of an asymptotic property is either zero or one.

  13. 13.

    In the case of the singular rules, the sequence must be outside the set of probability zero on which the rule produces an infinite subsequence; in the case of the nonsingular rules, it must be outside the set of probability zero on which the rule produces a subsequence that does not converge to 1/2.

  14. 14.

    Papers of Richard von Mises, HUG 4574.5 Box 3, Folder 1936, Harvard University Archives.

  15. 15.

    See [75, Lemma 1.6].

  16. 16.

    In the original: “nous nous soumettons en ce point”

  17. 17.

    Concerning early criticism of von Mises by Soviet philosophers, see Siegmund-Schultze [80].

  18. 18.

    In French: “La thèse de Monsieur Jean VILLE, intitulée Étude critique de la notion de Collectif, est une étude sur les fondements du calcul des probabilités, qui a eu les plus vifs éloges de Monsieur FRECHET et de Monsieur BOREL. Il est de fait que les assises de la théorie de la mesure étaient loin d’être clarifiées à l’époque où Jean VILLE a fait sa thèse, et que cette dernière a fortement contribué à sa mise au point.” (Archives Nationales, Fontainebleau, Cote 19840325, art. 542.)

  19. 19.

    This is corroborated by a letter Kolmogorov wrote to Fréchet in 1939; see the chapter “Andrei Kolmogorov and Leonid Levin on Randomness” in the present volume.

  20. 20.

    Many authors now use C(x) instead of K(x).

  21. 21.

    Zvonkin is listed as the first author of the article; note that Z comes before L in the Cyrillic alphabet.

  22. 22.

    Later Levin introduced the universal semimeasure on natural numbers; now it is sometimes called discrete a priori probability, while the universal semimeasure on a binary tree introduced in [94] is called continuous a priori probability.

  23. 23.

    Kolmogorov’s 1968 paper did not mention Levin explicitly, but the Zvonkin and Levin’s 1970 paper, carefully edited by Kolmogorov, mentioned that Levin and Kolmogorov independently came to the result, and there is no doubt that Kolmogorov agreed with this remark.

  24. 24.

    If understood literally, this definition does not have the properties mentioned by Levin in his letter; one should allow the output of A to appear bit by bit. The correct definition appeared in Levin’s paper [37].

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Acknowledgements

In addition to published sources, we have drawn on interviews with Peter Gács, Leonid Levin, and Per Martin-Löf, and on discussions at a meeting at Dagstuhl in late January and early February 2006. Marcus Hutter recorded historical talks at Dagstuhl by Christian Calude, Claus-Peter Schnorr, and Paul Vitányi and posted them at http://www.hutter1.net/dagstuhl. We have also profited from discussions with Leonid Bassalygo, Reinhard Siegmund-Schultze, Vladimir Uspensky, Vladimir Vovk, Vladimir Vyugin, and others. We are grateful to Reinhard Siegmund-Schultze for calling to our attention the letters between von Mises and Wald in the Harvard Archives.

Preparation of an earlier version of this chapter, which appeared in the Electronic Journal for History of Probability and Statistics in June 2009, was supported in part by ANR grant NAFIT-08-EMER-008-01. An earlier preliminary version [5], by Laurent Bienvenu and Alexander Shen, contains additional information about the history of algorithmic information theory; see also [77].

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  1. LaBRI (Laboratoire Bordelais de Recherche en Informatique), CNRS, Bordeaux, France

    Laurent Bienvenu

  2. Rutgers Business School, 1 Washington Park, Newark, New Jersey, USA

    Glenn Shafer

  3. LIRMM, University of Montpellier, CNRS, Montpellier, France

    Alexander Shen

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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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Bienvenu, L., Shafer, G., Shen, A. (2022). Martingales in the Study of Randomness. 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_11

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