On Convergence Rates of Spiked Eigenvalue Estimates: A General Study of Global and Local Laws in Sample Covariance Matrices

TL;DR

This paper studies the convergence rates of spiked eigenvalue estimates in sample covariance matrices, covering various growth regimes of matrix dimensions and establishing global and local spectral laws.

Contribution

It provides a comprehensive analysis of convergence rates for spiked eigenvalues under general growth conditions of matrix dimensions.

Findings

Derived convergence rates for spiked eigenvalues in diverse growth regimes.

Established global and local spectral laws for sample covariance matrices.

Applicable to high-dimensional settings with varying ratios of dimensions to sample size.

Abstract

This paper investigates global and local laws for sample covariance matrices with general growth rates of dimensions. The sample size $N$ and population dimension $M$ can have the same order in logarithm, which implies that their ratio $M/N$ can approach zero, a constant, or infinity. These theories are utilized to determine the convergence rate of spiked eigenvalue estimates.