  Multiple testing is a fundamental problem in high-dimensional statistical
inference. Although many methods have been proposed to control false
discoveries, it is still a challenging task when the tests are correlated to
each other. To overcome this challenge, various methods have been proposed to
estimate the false discovery rate (FDR) and/or the false discovery proportion
(FDP) under arbitrary covariance among the test statistics. An interesting
finding of these works is that the estimation of FDP and FDR under weak
dependence is identical to that under independence. However, Mei et al. (2021)
pointed out that unlike FDR, the asymptotic variance of FDP can still differ
drastically from that under independence, and the difference depends on the
covariance structure among the test statistics. In this paper, we further
extend this result from $z$-tests to $t$-tests when the marginal variances are
unknown and need to be estimated. With weakly dependent $t$-tests, we show that
FDP still converges to a fixed quantity unrelated to the dependence structure,
and further derive the asymptotic expansion and uncertainty of FDP leading to
similar results as in Mei et al. (2021). In addition, we develop an
approximation method to efficiently evaluate the asymptotic variance of FDP for
dependent $t$-tests. We examine how the asymptotic variance of FDP varies as
well as the performance of its estimators under different dependence structures
through simulations and a real-data study.
