  A general asymptotic framework is developed for studying consis- tency
properties of principal component analysis (PCA). Our frame- work includes
several previously studied domains of asymptotics as special cases and allows
one to investigate interesting connections and transitions among the various
domains. More importantly, it enables us to investigate asymptotic scenarios
that have not been considered before, and gain new insights into the
consistency, subspace consistency and strong inconsistency regions of PCA and
the boundaries among them. We also establish the corresponding convergence rate
within each region. Under general spike covariance models, the dimension (or
the number of variables) discourages the consistency of PCA, while the sample
size and spike information (the relative size of the population eigenvalues)
encourages PCA consistency. Our framework nicely illustrates the relationship
among these three types of information in terms of dimension, sample size and
spike size, and rigorously characterizes how their relationships affect PCA
consistency.
