  We consider a more generalized spiked covariance matrix $\Sigma$, which is a
general non-definite matrix with the spiked eigenvalues scattered into a few
bulks and the largest ones allowed to tend to infinity. By relaxing the
matching of the 4th moment to a tail probability decay, a {\it Generalized Four
Moment Theorem} (G4MT) is proposed to show the universality of the asymptotic
law for the local spectral statistics of generalized spiked covariance
matrices, which implies the limiting distribution of the spiked eigenvalues of
the generalized spiked covariance matrix is independent of the actual
distributions of the samples satisfying our relaxed assumptions. Moreover, by
applying it to the Central Limit Theorem (CLT) for the spiked eigenvalues of
the generalized spiked covariance matrix, we also extend the result of Bai and
Yao (2012) to a general form of the population covariance matrix, where the 4th
moment is not necessarily required to exist and the spiked eigenvalues are
allowed to be dependent on the non-spiked ones, thus meeting the actual cases
better.
