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

Consistent estimation of the proportion of false nulls and FDR for adaptive multiple testing Normal means under weak dependence