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 Probability Surveys > Vol. 17 (2020) open journal systems 


Time-uniform Chernoff bounds via nonnegative supermartingales

Steven R. Howard, University of California, Berkeley
Aaditya Ramdas, Carnegie Mellon University
Jon McAuliffe, University of California, Berkeley; The Voleon Group
Jasjeet Sekhon, University of California, Berkeley


Abstract
We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960–80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980–2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Peña; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cramér-Chernoff method, self-normalized processes, and other parts of the literature.

AMS 2000 subject classifications: Primary 60E15, 60G17; secondary 60F10, 60B20

Keywords: Exponential concentration inequalities; nonnegative supermartingale; line crossing probability

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Howard, Steven R., Ramdas, Aaditya, McAuliffe, Jon, Sekhon, Jasjeet, Time-uniform Chernoff bounds via nonnegative supermartingales, Probability Surveys, 17, (2020), 257-317 (electronic). DOI: 10.1214/18-PS321.

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