Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models

TL;DR

This paper establishes a joint central limit theorem for random vectors based on sesquilinear forms and applies it to large-dimensional spiked population models, revealing joint distributions of eigenvalues and eigenvector projections.

Contribution

It extends CLT theory to sesquilinear forms and applies it to analyze eigenvalues and eigenvectors in high-dimensional spiked models.

Findings

Joint distribution of grouped extreme eigenvalues derived

Asymptotic distribution of eigenvalue and eigenvector projection obtained

Extension of CLT to sesquilinear forms in high-dimensional settings

Abstract

In this paper, we derive a joint central limit theorem for random vector whose components are function of random sesquilinear forms. This result is a natural extension of the existing central limit theory on random quadratic forms. We also provide applications in random matrix theory related to large-dimensional spiked population models. For the first application, we find the joint distribution of grouped extreme sample eigenvalues correspond to the spikes. And for the second application, under the assumption that the population covariance matrix is diagonal with $k$ (fixed) simple spikes, we derive the asymptotic joint distribution of the extreme sample eigenvalue and its corresponding sample eigenvector projection.