A CLT for the LSS of large dimensional sample covariance matrices with diverging spikes

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded. This constitutes a nontrivial extension of the Bai-Silverstein theorem (BST) (Ann Probab 32(1):553--605, 2004), a theorem that has strongly influenced the development of high-dimensional statistics, especially in the applications of random matrix theory to statistics. Recently there has been a growing realization that the assumption of uniform boundedness of the population covariance matrices in BST is not satisfied in some fields, such as economics, where the variances of principal components could diverge as the dimension tends to infinity. Therefore, in this paper, we aim to eliminate the obstacles to the applications of BST. Our new CLT accommodates the spiked eigenvalues, which may either be bounded or tend to infinity. A distinguishing feature of our result is that the variance in the new CLT is related to both spiked eigenvalues and bulk eigenvalues, with dominance being determined by the divergence rate of the largest spiked eigenvalue. The new CLT for LSS is then applied to test the hypothesis that the population covariance matrix is the identity matrix or a generalized spiked model. The asymptotic distributions for the corrected likelihood ratio test statistic and corrected Nagao's trace test statistic are derived under the alternative hypothesis. Moreover, we provide power comparisons between the two LSSs and Roy's largest root test under certain hypotheses. In particular, we demonstrate that except for the case where the number of spikes is equal to 1, the LSSs may exhibit higher power than Roy's largest root test in certain scenarios.