Central limit theorems for eigenvalues in a spiked population model

TL;DR

This paper derives the limiting distributions and a central limit theorem for eigenvalues in a spiked population model, enhancing understanding of how population spikes influence sample eigenvalues in high-dimensional data.

Contribution

It establishes the limiting distributions of extreme sample eigenvalues and introduces a central limit theorem for random sesquilinear forms in the context of spiked models.

Findings

Limiting distributions of extreme sample eigenvalues are derived.

A central limit theorem for random sesquilinear forms is proved.

Results quantify the effect of population spikes on sample eigenvalues.

Abstract

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.