  We discuss several approaches to defining power in studies designed around
the Benjamini-Hochberg (BH) false discovery rate (FDR) procedure. We focus
primarily on the \textit{average power} and the $\lambda$-\textit{power}, which
are the expected true positive fraction and the probability that the true
positive fraction exceeds $\lambda$, respectively. We prove results concerning
strong consistency and asymptotic normality for the positive call fraction
(PCF), the true positive fraction (TPF) and false discovery fraction (FDF).
Convergence of their corresponding expected values, including a convergence
result for the average power, follow as a corollaries. After reviewing what is
known about convergence in distribution of the errors of the plugin procedure,
(Genovese, 2004), we prove central limit theorems for fully empirical versions
of the PCF, TPF, and FDF, using a result for stopped stochastic processes. The
central limit theorem (CLT) for the TPF is used to obtain an approximate
expression for the $\lambda$-power, while the CLT for the FDF is used to
introduce an approximate procedure for determining a suitably small nominal FDR
that results in a speicified bound on the FDF with stipulated high probability.
The paper also contains the results of a large simulation study covering a
fairly substantial portion of the space of possible inputs encountered in
application of the results in the design of a biomarker study, a micro-array
experiment and a GWAS study.
