Asymptotics of ultra-high-dimensional generalized spiked sample covariance matrix

TL;DR

This paper studies the asymptotic behavior of eigenvalues and eigenvectors of sample covariance matrices in ultra-high-dimensional settings where the dimension grows faster than the sample size.

Contribution

It establishes the first-order convergence limits of eigenvalues and eigenvectors in ultra-high-dimensional spiked covariance models, extending previous results.

Findings

Derived convergence limits for eigenvalues.

Derived convergence limits for eigenvector projections.

Applicable to ultra-high-dimensional regimes where p grows faster than n.

Abstract

This paper investigates the asymptotics of eigenstructure of sample covariance matrix under the spiked covariance matrix model in ultra-high-dimensional settings, where the dimensionality can grow much faster than the sample size with $ p \asymp n^{\alpha} $, $ \alpha > 1 $. We establish the first-order convergence limits of eigenvalue locations and eigenvector projections of properly scaled sample covariance matrix. Our results are extensions of \cite{bloemendal16,ding21}.