More powerful multiple testing under dependence via randomization

We develop a technique to improve the power of any e-value by a simple randomization involving one independent uniform random variable. Using this framework, 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 be improved through randomization. We also improve the Hommel test under dependence, and post-selection inference procedures for confidence intervals with false coverage rate (FCR) that allow for arbitrary selection rules and dependence. Importantly, our randomized improvements are never worse than the originals and are typically strictly more powerful, with marked improvements in simulations.