  We propose sequential multiple testing procedures which control the false
discover rate (FDR) or the positive false discovery rate (pFDR) under arbitrary
dependence between the data streams. This is accomplished by "optimizing" an
upper bound on these error metrics for a class of step down sequential testing
procedures. Both open-ended and truncated versions of these sequential
procedures are given, both being able to control both the type~I multiple
testing metric (FDR or pFDR) at specified levels, and the former being able to
control both the type I and type II (e.g., FDR and the false nondiscovery rate,
FNR). In simulation studies, these procedures provide 45-65% savings in average
sample size over their fixed-sample competitors. We illustrate our procedures
on drug data from the United Kingdom's Yellow Card Pharmacovigilance Database.
