Central limit theorems for eigenvalues in a spiked population model

Details Central limit theorems for eigenvalues in a spiked population model Central limit theorems for eigenvalues in a spiked population model

2008-06-16

arxiv.org/abs/0806.2503v1

Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 3, 447-474

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AIHP118 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu

Scientific paper

10.1214/07-AIHP118

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.

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