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Encounters with Martingales in Statistics and Stochastic Optimization

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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

In this intellectual memoir, Professor Lai describes how martingales came into his world of mathematical statistics, first in sequential tests and confidence intervals, then in time series, stochastic approximation, sequential design of experiments, and stochastic optimization. He sketches the trajectories of many other statisticians that he met along the way. He emphasizes the roles of Harold Hotelling, Abraham Wald, and Herbert Robbins in their creation of the environment for the study of martingales at Columbia University and then his own subsequent work at Stanford University. At Stanford, he came to see stochastic optimization as a unifying theme for the use of martingales in statistical modeling. In conclusion, he describes the multifaceted applications of martingales to statistics and stochastic optimization in the BigData and MultiCloud era.

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Notes

  1. 1.

    See [65] and the chapter by Laurent Bienvenu, Glenn Shafer, and Alexander Shen in the present volume.

  2. 2.

    See Bernard Locker’s chapter in the present volume.

  3. 3.

    A revision of this 2009 article on martingales in survival analysis, by Aalen, Andersen, Borgan, Gill and Keiding, appears as a chapter in the present volume.

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Acknowledgements

This chapter expands my 2009 review of the history martingales in the Electronic Journal for History of Probability and Statistics [47] by expanding the scope from “Sequential Analysis and Time Series” to “Statistics and Stochastic Optimization” and expanding the period covered from 1945–1985 to 1933–2021. The research was supported by National Science Foundation grant DMS-1811818.

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  1. Department of Statistics, Stanford University, California, 94305, USA

    Tze Leung Lai

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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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Lai, T.L. (2022). Encounters with Martingales in Statistics and Stochastic Optimization. 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_12

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