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hal-00284468, version 1

Limit theorems for sample eigenvalues in a generalized spiked population model

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

Abstract: 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.

  • 1:  Key Laboratory of Applied Statistics under Ministry of Education (KLASMOE)
  • Northeast Normal University
  • 2:  Department of Statistics and Applied Probability (DSAP)
  • National University of Singapore
  • 3:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • Domain : Mathematics/Statistics
    Statistics/Statistics Theory
    Mathematics/Probability
  • Keywords : Sample covariance matrices – Spiked population model – Central limit theorems – Largest eigenvalue – Extreme eigenvalues.
  • Comment : 24 pages – 4 figures
  • Available versions :  v1 (2008-06-03) v2 (2008-06-06)
 
  • hal-00284468, version 1
  • http://hal.archives-ouvertes.fr/hal-00284468
  • oai:hal.archives-ouvertes.fr:hal-00284468
  • From: Jian-Feng Yao <>
  • Submitted on: Tuesday, 3 June 2008 10:43:12
  • Updated on: Tuesday, 3 June 2008 11:11:11