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

arXiv: https://arxiv.org/abs/1808.05362 · HTML: https://mlsanity.com/paper/1808.05362/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2018.

## Built on by (most cited first, 3 of 3)

- [Spectrally-Corrected and Regularized Linear Discriminant Analysis for Spiked Covariance Model](https://mlsanity.com/paper/2210.03859/) (2022)
- [Structural Effect and Spectral Enhancement of High-Dimensional Regularized Linear Discriminant Analysis](https://mlsanity.com/paper/2507.16682/) (2025)
- [High-dimensional ridgeless least squares interpolation under spiked covariance structures](https://mlsanity.com/paper/2608.07281/) (2026)

## Abstract (arXiv)

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.