ASYMPTOTICS OF EIGENSTRUCTURE OF SAMPLE CORRELATION MATRICES FOR HIGH-DIMENSIONAL SPIKED MODELS

Sample correlation matrices are widely used, but for high-dimensional data little is known about their spectral properties beyond null models , which assume the data have independent coordinates. In the class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both leading eigenvalues and eigenvectors of sample correlation matrices, assuming a high-dimensional regime in which the ratio p/n, of number of variables p to sample size n, converges to a positive constant. While the first-order spectral properties of sample correlation matrices match those of sample covariance matrices, their asymptotic distributions can differ significantly. Indeed, the correlation-based fluctuations of both sample eigenvalues and eigenvectors are often remarkably smaller than those of their sample covariance counterparts.

Automated summary

For high-dimensional data, using random matrix theory can help derive asymptotic results for the spectral properties of sample correlation matrices. While the first-order spectral properties of sample correlation matrices match those of sample covariance matrices, their asymptotic distributions can differ significantly. The fluctuations of both sample eigenvalues and eigenvectors based on correlations are often much smaller than those of their sample covariance counterparts.