Some Asymptotic Results on Multiple Testing under Weak Dependence

This paper studies the means-testing problem under weakly correlated Normal setups. Although quite common in genomic applications, test procedures having exact FWER control under such dependence structures are nonexistent. We explore the asymptotic behaviors of the classical Bonferroni (when adjusted suitably) and the Sidak procedure; and show that both of these control FWER at the desired level exactly as the number of hypotheses approaches infinity. We derive analogous limiting results on the generalized family-wise error rate and power. Simulation studies depict the asymptotic exactness of the procedures empirically.