A game-theoretic reformulation of sequential detection shows the LIL as the minimax boundary, with the optimal mixing prior being the Jeffreys prior on the scale-of-scales selected by the Erdős-Kolmogorov test, yielding a 3/2 coefficient for the first iterated-log correction.
Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon
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The multiply iterated law of the iterated logarithm: game-theoretic foundations of sequential detection boundaries
A game-theoretic reformulation of sequential detection shows the LIL as the minimax boundary, with the optimal mixing prior being the Jeffreys prior on the scale-of-scales selected by the Erdős-Kolmogorov test, yielding a 3/2 coefficient for the first iterated-log correction.