Limit theorems for sample eigenvalues in a generalized spiked population model

Zhidong Bai (1, 2) , Jian-Feng Yao (3)

1 KLASMOE - Key Laboratory of Applied Statistics under Ministry of Education
2 DSAP - Department of Statistics and Applied Probability
3 IRMAR - Institut de Recherche Mathématique de Rennes

In the spiked population model introduced by Johnstone (2001),the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work (Bai and Yao, 2008), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a {\em generalized} spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes.

Mots clés: Extreme eigenvalues; Sample covariance matrices; Spiked population model; Central limit theorems; Largest eigenvalue; Extreme eigenvalues.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Probabilités [math.PR]

Zhidong Bai, Jian-Feng Yao. Limit theorems for sample eigenvalues in a generalized spiked population model. 2008. ⟨hal-00284468v2⟩