Partial generalized four moment theorem revisited

Abstract

This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption ${max}_{\mathit{t},\mathit{s}}|{\mathit{u}}_{\mathit{t}\mathit{s}}{|}^2\mathrm{E}\{|{\mathit{x}}_{11}{|}^4\mathit{I}(|{\mathit{x}}_{11}|<\sqrt{\mathit{n}})-\mathit{\mu }\}\to 0$ with the same μ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in Bai and Yao (J. Multivariate Anal. 106 (2012) 167–177) as a special case.