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.

翻译：本文建立了当总体协方差矩阵非一致有界时，大维样本协方差矩阵线性谱统计量（LSS）的中心极限定理（CLT）。这构成了 Bai-Silverstein 定理（BST）（Ann Probab 32(1):553--605, 2004）的重要推广。BST 定理深刻影响了高维统计学的发展，尤其是在随机矩阵理论应用于统计学领域方面。近期人们日益认识到，在某些领域（如经济学），BST 定理中总体协方差矩阵的一致有界性假设并不成立——在这些领域中，主成分的方差可能随维度趋于无穷而发散。因此，本文旨在消除 BST 定理的应用障碍。我们提出的新中心极限定理适用于尖峰特征值，这些特征值可能有界也可能趋于无穷。该结果的一个显著特征是，新中心极限定理中的方差同时涉及尖峰特征值与体特征值，其主导地位由最大尖峰特征值的发散速率决定。随后，我们将新中心极限定理应用于检验总体协方差矩阵为单位矩阵或广义尖峰模型的假设。在备择假设下，推导了校正似然比检验统计量与校正长尾迹检验统计量的渐近分布。此外，我们还在特定假设下比较了两种线性谱统计量与 Roy 最大根检验的功效。特别地，除尖峰数量等于1的情况外，我们证明线性谱统计量在某些场景下可能表现出比 Roy 最大根检验更高的检验功效。