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Central limit theorems for eigenvalues in a spiked population model. (English) Zbl 1274.62129
Summary: In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.

62E20 Asymptotic distribution theory in statistics
62H25 Factor analysis and principal components; correspondence analysis
60F05 Central limit and other weak theorems
60B20 Random matrices (probabilistic aspects)
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