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


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.


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Steven R. Howard. Aaditya Ramdas. Jon McAuliffe. Jasjeet Sekhon. "Time-uniform Chernoff bounds via nonnegative supermartingales." Probab. Surveys 17 257 - 317, 2020.


Received: 1 November 2018; Published: 2020
First available in Project Euclid: 20 May 2020

Digital Object Identifier: 10.1214/18-PS321

Primary: 60E15 , 60G17
Secondary: 60B20 , 60F10

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

Vol.17 • 2020
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