  Stepwise multiple testing procedures have attracted several statisticians for
decades and are also quite popular with statistics users because of their
technical simplicity. The Bonferroni procedure has been one of the earliest and
most prominent testing rules for controlling the familywise error rate (FWER).
A recent article established that the FWER for the Bonferroni method
asymptotically (i.e., when the number of hypotheses becomes arbitrarily large)
approaches zero under any positively equicorrelated multivariate normal
framework. However, similar results for the limiting behaviors of FWER of
general stepwise procedures are nonexistent. The present work addresses this
gap in a unified manner by studying the limiting behaviors of the FWER of
several stepwise testing rules for correlated normal setups. Specifically, we
show that the limiting FWER approaches zero for any step-down rule (e.g.,
Holm's method) provided the infimum of the correlations is strictly positive.
We also establish similar limiting zero results on FWER of other popular
multiple testing rules, e.g., Hochberg's and Hommel's procedures. We then
extend these results to any configuration of true and false null hypotheses. It
turns out that, within our chosen asymptotic framework, the Benjamini-Hochberg
method can hold the FWER at a strictly positive level asymptotically under the
equicorrelated normality. We finally discuss the limiting powers of various
procedures.
