Generalized Four Moment Theorem with an application to the CLT for the spiked eigenvalues of high-dimensional general Fisher-matrices

TL;DR

This paper introduces a generalized four moment theorem for high-dimensional spiked Fisher matrices, relaxing previous restrictions and establishing a broader universality result for the distribution of spiked eigenvalues.

Contribution

It proposes a new G4MT that reduces moment matching constraints to tail probabilities, extending universality to more general Fisher matrices without bounded fourth moments.

Findings

The limiting distribution of spiked eigenvalues is distribution-free under relaxed assumptions.

The G4MT applies to a wider class of Fisher matrices, including non-diagonal and unbounded moment cases.

Extension of CLT for spiked eigenvalues beyond previous diagonal block independence constraints.

Abstract

The universality for the local spiked eigenvalues is a powerful tool to deal with the problems of the asymptotic law for the bulks of spiked eigenvalues of high-dimensional generalized Fisher matrices. In this paper, we focus on a more generalized spiked Fisher matrix, where $\Sigma_1\Sigma_2^{-1}$ is free of the restriction of diagonal independence, and both of the spiked eigenvalues and the population 4th moments are not necessary required to be bounded. By reducing the matching four moments constraint to a tail probability, we propose a Generalized Four Moment Theorem (G4MT) for the bulks of spiked eigenvalues of high-dimensional generalized Fisher matrices, which shows that the limiting distribution of the spiked eigenvalues of a generalized spiked Fisher matrix is independent of the actual distributions of the samples provided to satisfy the our relaxed assumptions. Furthermore, as an illustration, we also apply the G4MT to the Central Limit Theorem for the spiked eigenvalues of generalized spiked Fisher matrix, which removes the strict condition of the diagonal block independence given in Wang and Yao (2017) and extends their result to a wider usage without the requirements of the bounded 4th moments and the diagonal block independent structure, meeting the actual cases better.