  Testing independence among a number of (ultra) high-dimensional random
samples is a fundamental and challenging problem. By arranging $n$ identically
distributed $p$-dimensional random vectors into a $p \times n$ data matrix, we
investigate the problem of testing independence among columns under the
matrix-variate normal modeling of data. We propose a computationally simple and
tuning-free test statistic, characterize its limiting null distribution,
analyze the statistical power and prove its minimax optimality. As an important
by-product of the test statistic, a ratio-consistent estimator for the
quadratic functional of a covariance matrix from correlated samples is
developed. We further study the effect of correlation among samples to an
important high-dimensional inference problem --- large-scale multiple testing
of Pearson's correlation coefficients. Indeed, blindly using classical
inference results based on the assumed independence of samples will lead to
many false discoveries, which suggests the need for conducting independence
testing before applying existing methods. To address the challenge arising from
correlation among samples, we propose a "sandwich estimator" of Pearson's
correlation coefficient by de-correlating the samples. Based on this approach,
the resulting multiple testing procedure asymptotically controls the overall
false discovery rate at the nominal level while maintaining good statistical
power. Both simulated and real data experiments are carried out to demonstrate
the advantages of the proposed methods.
