  We consider multiple testing means of many dependent Normal random variables
that do not necessarily follow a joint Normal distribution. Under weak
dependence, we show the uniform consistency of proportion estimators that are
constructed as solutions to Lebesgue-Stieltjes equations for the setting of a
point, bounded and one-sided null, respectively, and characterize via the index
of weak dependence the sparsest proportion these estimators can consistently
estimate. On the other hand, under a principal correlation structure and
employing a suitable definition of p-value for composite null hypotheses, we
show that three key empirical processes induced by a single-step multiple
testing procedure (MTP) satisfy the strong law of large numbers for testing
each of the three types of nulls. Further, under this structure and for testing
a point null and a one-sided null respectively, we construct an adaptive
single-step MTP that employs a proportion estimator mentioned earlier, and show
that the false discovery proportion of this procedure satisfies the weak law of
large numbers and hence consistently estimates the false discovery rate of the
procedure. In addition, we report some findings on the estimators of Jin and of
Meinshausen and Rice of the proportion of false nulls in the critically and
very sparse regimes under weak dependence and model misspecifications,
respectively.
