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In this paper, we establish the central limit theorem (CLT) for linearspectral statistics (LSS) of large-dimensional sample covariance matrix whenthe population covariance matrices are not uniformly bounded, which is anontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latterhas strongly stimulated the development of high-dimensional statistics,especially the application of random matrix theory to statistics. However, theassumption of uniform boundedness of the population covariance matrices isfound strongly limited to the applications of BST. The aim of this paper is toremove the blockages to the applications of BST. The new CLT, allows the spikedeigenvalues to exist and tend to infinity. It is interesting to note that theroles of either spiked eigenvalues or the bulk eigenvalues or both of the twoare dominating in the CLT.  Moreover, the results are checked by simulation studies with variouspopulation settings. The CLT for LSS is then applied for testing the hypothesisthat a covariance matrix $ \bSi $ is equal to an identity matrix. For this, theasymptotic distributions for the corrected likelihood ratio test (LRT) andNagao's trace test (NT) under alternative are derived, and we also propose theasymptotic power of LRT and NT under certain alternatives.

A CLT for the LSS of large dimensional sample covariance matrices with unbounded dispersions