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The universality for the local spiked eigenvalues is a powerful tool to dealwith the problems of the asymptotic law for the bulks of spiked eigenvalues ofhigh-dimensional generalized Fisher matrices. In this paper, we focus on a moregeneralized spiked Fisher matrix, where $\Sigma_1\Sigma_2^{-1}$ is free of therestriction of diagonal independence, and both of the spiked eigenvalues andthe population 4th moments are not necessary required to be bounded. Byreducing the matching four moments constraint to a tail probability, we proposea Generalized Four Moment Theorem (G4MT) for the bulks of spiked eigenvalues ofhigh-dimensional generalized Fisher matrices, which shows that the limitingdistribution of the spiked eigenvalues of a generalized spiked Fisher matrix isindependent of the actual distributions of the samples provided to satisfy theour relaxed assumptions. Furthermore, as an illustration, we also apply theG4MT to the Central Limit Theorem for the spiked eigenvalues of generalizedspiked Fisher matrix, which removes the strict condition of the diagonal blockindependence given in Wang and Yao (2017) and extends their result to a widerusage without the requirements of the bounded 4th moments and the diagonalblock independent structure, meeting the actual cases better.

Generalized Four Moment Theorem with an application to the CLT for the spiked eigenvalues of high-dimensional general Fisher-matrices