CLT for spiked eigenvalues of a sample covariance matrix from high-dimensional Gaussian mean mixtures

Abstract

Sample covariance matrices from a finite mean mixture model naturally carry certain spiked eigenvalues, which are generated by the differences among the mean vectors. However, their asymptotic behaviors remain largely unknown when the population dimension p grows proportionally to the sample size n. In this paper, a new CLT is established for the spiked eigenvalues by considering a Gaussian mean mixture in such high-dimensional asymptotic frameworks. It shows that the convergence rate of these eigenvalues is O(1/n) and their fluctuations can be characterized by the mixing proportions, the eigenvalues of the common covariance matrix, and the inner products between the mean vectors and the eigenvectors of the covariance matrix.