  The aim of this paper is to establish several deep theoretical properties of
principal component analysis for multiple-component spike covariance models.
Our new results reveal a surprising asymptotic conical structure in critical
sample eigendirections under the spike models with distinguishable (or
indistinguishable) eigenvalues, when the sample size and/or the number of
variables (or dimension) tend to infinity. The consistency of the sample
eigenvectors relative to their population counterparts is determined by the
ratio between the dimension and the product of the sample size with the spike
size. When this ratio converges to a nonzero constant, the sample eigenvector
converges to a cone, with a certain angle to its corresponding population
eigenvector.In the High Dimension, Low Sample Size case, the angle between the
sample eigenvector and its population counterpart converges to a limiting
distribution.Several generalizations of the multi-spike covariance models are
also explored, and additional theoretical results are presented.
