Central limit theorems for eigenvalues in a 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 a spiked population model, the population covariance matrix has all its eigenvalues equal to unit 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.

Mots clés: Sample covariance matrices; Spiked population model; Central limit theorems; Largest eigenvalue; Extreme eigenvalues; Random sesquilinear forms; Random quadratic forms

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

Zhidong Bai, Jian-Feng Yao. Central limit theorems for eigenvalues in a spiked population model. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2008, 44 (3), pp.447-474. ⟨hal-00129331⟩