  We derive the asymptotic distributions of the spiked eigenvalues and
eigenvectors under a generalized and unified asymptotic regime, which takes
into account the spike magnitude of leading eigenvalues, sample size, and
dimensionality. This new regime allows high dimensionality and diverging
eigenvalue spikes and provides new insights into the roles the leading
eigenvalues, sample size, and dimensionality play in principal component
analysis. The results are proven by a technical device, which swaps the role of
rows and columns and converts the high-dimensional problems into
low-dimensional ones. Our results are a natural extension of those in Paul
(2007) to more general setting with new insights and solve the rates of
convergence problems in Shen et al. (2013). They also reveal the biases of the
estimation of leading eigenvalues and eigenvectors by using principal component
analysis, and lead to a new covariance estimator for the approximate factor
model, called shrinkage principal orthogonal complement thresholding (S-POET),
that corrects the biases. Our results are successfully applied to outstanding
problems in estimation of risks of large portfolios and false discovery
proportions for dependent test statistics and are illustrated by simulation
studies.
