  Assessing the stability of a multiple testing procedure under dependence is
important but very challenging. Even for multiple testing which among a set of
Normal random variables have mean zero, which we refer to as the "Normal means
problem", to date there lacks a classification of the type of dependence under
which the strong law of large numbers (SLLN) holds for the numbers of
rejections and false rejections. We introduce the concept of "principal
correlation structure (PCS)" that characterizes the type of dependence for
which such SLLN holds, and establish the law. Further, we show that PCS ensures
the SLLN for the false discover proportion when there is always a positive
proportion of zero Normal means. We also investigate the stability of two
conditional multiple testing procedures for the Normal means problem, and show
that the associated SLLN holds when in addition the decomposition of the
covariance matrix of the Normal random variables that induces PCS is
homogeneous in certain sense. Our results also provide a formal way to check if
the "weak dependence" assumption, a widely used assumption in the multiple
testing literature, holds for the Normal means problem. As by-products, we
establish a universal bound on Hermite polynomials and a universal comparison
result on the covariance of the indicator functions of the two p-values of
testing the marginal means of a bivariate Normal random vector and the
correlation between the two components of the vector. These are of their own
interests.
