Limit theorems for sample eigenvalues in a generalized spiked population model

TL;DR

This paper extends the theory of sample eigenvalues in spiked population models to a generalized setting with arbitrary base covariance, providing new convergence results and mathematical tools.

Contribution

It introduces a generalized spiked population model and develops new methods to analyze the convergence of sample eigenvalues caused by spikes.

Findings

Established almost sure convergence of sample eigenvalues in the generalized model

Extended limiting distribution results to arbitrary base covariance matrices

Provided new mathematical tools for analyzing eigenvalue perturbations

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.