On the tradeoff between almost sure error tolerance and its mean deviation frequency in martingale convergence

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 modulus of convergence. For this purpose we generalize an elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies, which was recently established by the authors. First we study martingale convergence in L2, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences. Applications are the tradeoff for the multicolor generalized Pólya urn processes, the generalized Chinese restaurant process, statistical M-estimators, as well as excursion frequencies of the Galton-Watson branching process.