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We consider a more generalized spiked covariance matrix $\Sigma$, which is ageneral non-definite matrix with the spiked eigenvalues scattered into a fewbulks and the largest ones allowed to tend to infinity. By relaxing thematching of the 4th moment to a tail probability decay, a {\it Generalized FourMoment Theorem} (G4MT) is proposed to show the universality of the asymptoticlaw for the local spectral statistics of generalized spiked covariancematrices, which implies the limiting distribution of the spiked eigenvalues ofthe generalized spiked covariance matrix is independent of the actualdistributions of the samples satisfying our relaxed assumptions. Moreover, byapplying it to the Central Limit Theorem (CLT) for the spiked eigenvalues ofthe generalized spiked covariance matrix, we also extend the result of Bai andYao (2012) to a general form of the population covariance matrix, where the 4thmoment is not necessarily required to exist and the spiked eigenvalues areallowed to be dependent on the non-spiked ones, thus meeting the actual casesbetter.

Generalized Four Moment Theorem and an Application to CLT for Spiked Eigenvalues of Large-dimensional Covariance Matrices