Generalized four moment theorem and an application to CLT for spiked eigenvalues of high-dimensional covariance matrices

Abstract

We consider a more generalized spiked covariance matrix, which is a general non-negative definite matrix with the spiked eigenvalues scattered into spaces of a few bulks and the largest ones allowed to tend to infinity. The study is split into two cases by whether the maximum absolute value of the eigenvector of the corresponding spikes tends to zero or not. On one hand, if it is zero, a Generalized Four Moment Theorem (G4MT) is proposed by relaxing the matching of the 3rd and the 4th moment to the tail probability decaying with certain rate, which shows the universality of the asymptotic law for the spiked eigenvalues of the generalized spiked covariance model. On the other hand, if it is not zero, the matches of the third and fourth moments in usual four moment theorem are weakened to only requiring the match of the 4th moment. Moreover, by applying the results to the Central Limit Theorem (CLT) for the spiked eigenvalues of the generalized spiked covariance model, we successively remove the restrictive condition of block wise diagonal assumption on the population covariance matrix in the previous works. This condition implies an unrealistic fact that the spiked eigenvalues and bulked eigenvalues are generated by independent variables, respectively. Thus, the new CLT will have much better application domain.