  We investigate asymptotically optimal multiple testing procedures for streams
of sequential data in the context of prior information on the number of false
null hypotheses ("signals"). We show that the "gap" and "gap-intersection"
procedures, recently proposed and shown by Song and Fellouris (2017, Electron.
J. Statist.) to be asymptotically optimal for controlling type 1 and 2
familywise error rates (FWEs), are also asymptotically optimal for controlling
FDR/FNR when their critical values are appropriately adjusted. Generalizing
this result, we show that these procedures, again with appropriately adjusted
critical values, are asymptotically optimal for controlling any multiple
testing error metric that is bounded between multiples of FWE in a certain
sense. This class of metrics includes FDR/FNR but also pFDR/pFNR, the
per-comparison and per-family error rates, and the false positive rate. Our
analysis includes asymptotic regimes in which the number of null hypotheses
approaches $\infty$ as the type 1 and 2 error metrics approach $0$.
