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- W4387687841 abstract "In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective statistics of the deviation frequencies (error incidence). For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma, which was recently established by the authors. First we study martingale convergence in $L^2$, and in the settings of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $pgeq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Voln'y. We also present applications for the tradeoff for the multicolor generalized P'olya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s. excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric." @default.
- W4387687841 created "2023-10-17" @default.
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- W4387687841 date "2023-10-13" @default.
- W4387687841 modified "2023-10-18" @default.
- W4387687841 title "On the tradeoff between almost sure error tolerance versus mean deviation frequency in martingale convergence" @default.
- W4387687841 doi "https://doi.org/10.48550/arxiv.2310.09055" @default.
- W4387687841 hasPublicationYear "2023" @default.
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