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We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative martingale which explodes to infinity when that event occurs.
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Kenshi Miyabe and Akimichi Takemura, The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges , Stochastic Process. Appl. 123
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Glenn Shafer, Testing by betting: a strategy for statistical and scientific communication , J. Roy. Statist. Soc. Ser. A 184
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Wouter M. Koolen and Peter Grünwald, Log-optimal anytime-valid E-values , Internat. J. Approx. Reason. 141
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Laurent Mazliak and Glenn Shafer (eds.), The Splendors and Miseries of Martingales—Their History from the Casino to Mathematics , Trends in the History of Science, Birkhäuser/Springer, Cham, [2022] ©2022. MR 4516531
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Hans Föllmer and Alexander Schied, Stochastic finance , De Gruyter Graduate, De Gruyter, Berlin, 2016, An introduction in discrete time, Fourth revised and extended edition of [ MR1925197]. MR 3859905
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Takeyuki Sasai, Kenshi Miyabe, and Akimichi Takemura, Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach , Ann. Probab. 47
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Peter Grünwald, Rianne de Heide, and Wouter M. Koolen, Safe testing , Journal of the Royal Statistical Society, Series B (to appear) (2023)
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J. L. Doob, Regularity properties of certain families of chance variables , Trans. Amer. Math. Soc. 47
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