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

Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions