  Online testing procedures assume that hypotheses are observed in sequence,
and allow the significance thresholds for upcoming tests to depend on the test
statistics observed so far. Some of the most popular online methods include
alpha investing, LORD++ (hereafter, LORD), and SAFFRON. These three methods
have been shown to provide online control of the "modified" false discovery
rate (mFDR) under a condition known as conditional superuniformity. However, to
our knowledge, LORD & SAFFRON have only been shown to control the traditional
false discovery rate (FDR) under an independence condition on the test
statistics. Our work bolsters these results by showing that SAFFRON and LORD
additionally ensure online control of the FDR under a "local" form of
nonnegative dependence. Further, FDR control is maintained under certain types
of adaptive stopping rules, such as stopping after a certain number of
rejections have been observed. Because alpha investing can be recovered as a
special case of the SAFFRON framework, our results immediately apply to alpha
investing as well. In the process of deriving these results, we also formally
characterize how the conditional superuniformity assumption implicitly limits
the allowed p-value dependencies. This implicit limitation is important not
only to our proposed FDR result, but also to many existing mFDR results.
