Generalized Four Moment Theorem and an Application to CLT for Spiked Eigenvalues of Large-dimensional Covariance Matrices

TL;DR

This paper introduces a generalized universality theorem for the spectral behavior of spiked covariance matrices, relaxing moment conditions and extending CLT results to more general, possibly dependent, eigenvalue structures.

Contribution

It proposes the Generalized Four Moment Theorem (G4MT) that broadens the universality of spectral statistics for generalized spiked covariance matrices with relaxed assumptions.

Findings

Universality of spectral statistics under relaxed moment conditions

Extension of CLT for spiked eigenvalues to dependent and non-4th-moment cases

Applicability to more general covariance matrix structures

Abstract

We consider a more generalized spiked covariance matrix $\Sigma$, which is a general non-definite matrix with the spiked eigenvalues scattered into a few bulks and the largest ones allowed to tend to infinity. By relaxing the matching of the 4th moment to a tail probability decay, a {\it Generalized Four Moment Theorem} (G4MT) is proposed to show the universality of the asymptotic law for the local spectral statistics of generalized spiked covariance matrices, which implies the limiting distribution of the spiked eigenvalues of the generalized spiked covariance matrix is independent of the actual distributions of the samples satisfying our relaxed assumptions. Moreover, by applying it to the Central Limit Theorem (CLT) for the spiked eigenvalues of the generalized spiked covariance matrix, we also extend the result of Bai and Yao (2012) to a general form of the population covariance matrix, where the 4th moment is not necessarily required to exist and the spiked eigenvalues are allowed to be dependent on the non-spiked ones, thus meeting the actual cases better.