  We investigate the performance of a family of multiple comparison procedures
for strong control of the False Discovery Rate ($\mathsf{FDR}$). The
$\mathsf{FDR}$ is the expected False Discovery Proportion ($\mathsf{FDP}$),
that is, the expected fraction of false rejections among all rejected
hypotheses. A number of refinements to the original Benjamini-Hochberg
procedure [1] have been proposed, to increase power by estimating the
proportion of true null hypotheses, either implicitly, leading to one-stage
adaptive procedures [4, 7] or explicitly, leading to two-stage adaptive (or
plug-in) procedures [2, 21]. We use a variant of the stochastic process
approach proposed by Genovese and Wasserman [11] to study the fluctuations of
the $\mathsf{FDP}$ achieved with each of these procedures around its
expectation, for independent tested hypotheses. We introduce a framework for
the derivation of generic Central Limit Theorems for the $\mathsf{FDP}$ of
these procedures, characterizing the associated regularity conditions, and
comparing the asymptotic power of the various procedures. We interpret recently
proposed one-stage adaptive procedures [4, 7] as fixed points in the iteration
of well known two-stage adaptive procedures [2, 21].
