We study the asymptotic distributions of the spiked eigenvalues and thelargest nonspiked eigenvalue of the sample covariance matrix under a generalcovariance matrix model with divergent spiked eigenvalues, while the othereigenvalues are bounded but otherwise arbitrary. The limiting normaldistribution for the spiked sample eigenvalues is established. It has distinctfeatures that the asymptotic mean relies on not only the population spikes butalso the nonspikes and that the asymptotic variance in general depends on thepopulation eigenvectors. In addition, the limiting Tracy-Widom law for thelargest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leadingeigenvectors are also considered. The results hold even when the number of thespikes diverges. As a key technical tool, we develop a Central Limit Theoremfor a type of random quadratic forms where the random vectors and randommatrices involved are dependent. This result can be of independent interest.
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