  In online multiple testing, an a priori unknown number of hypotheses are
tested sequentially, i.e. at each time point a test decision for the current
hypothesis has to be made using only the data available so far. Although many
powerful test procedures have been developed for online error control in recent
years, most of them are designed solely for independent or at most locally
dependent test statistics. In this work, we provide a new framework for
deriving online multiple test procedures which ensure asymptotical (with
respect to the sample size) control of the familywise error rate (FWER),
regardless of the dependence structure between test statistics. In this
context, we give a few concrete examples of such test procedures and discuss
their properties. Furthermore, we conduct a simulation study in which the type
I error control of these test procedures is also confirmed for a finite sample
size and a gain in power is indicated.
