Limiting Laws for Divergent Spiked Eigenvalues and Largest Non-spiked Eigenvalue of Sample Covariance Matrices

Abstract

We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy-Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a Central Limit Theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.

References in corpus (1)

• Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions

Cited by in corpus (3)

• Estimating Number of Factors by Adjusted Eigenvalues Thresholding
• The limits of the sample spiked eigenvalues for a high-dimensional generalized Fisher matrix and its applications
• Generalized Four Moment Theorem with an application to the CLT for the spiked eigenvalues of high-dimensional general Fisher-matrices