  We show that two procedures for false discovery rate (FDR) control -- the
Benjamini-Yekutieli procedure for dependent p-values, and the
e-Benjamini-Hochberg procedure for dependent e-values -- can both be made more
powerful by a simple randomization involving one independent uniform random
variable. As a corollary, the Hommel test under arbitrary dependence is also
improved. Importantly, our randomized improvements are never worse than the
originals and are typically strictly more powerful, with marked improvements in
simulations. The same technique also improves essentially every other multiple
testing procedure based on e-values.
