  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 (error incidence), 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 $L^p$, $p\geq 2$, 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.
