In the case where the dimension of the data grows at the same rate as the sample size we prove a central limit theorem for the difference of a linear spectral statistic of the sample covariance and a linear spectral statistic of the matrix that is obtained from the sample covariance matrix by deleting a column and the corresponding row. Unlike previous works, we do neither require that the population covariance matrix is diagonal nor that moments of all order exist. Our proof methodology incorporates subtle enhancements to existing strategies, which meet the challenges introduced by determining the mean and covariance structure for the difference of two such eigenvalue statistics. Moreover, we also establish the asymptotic independence of the difference-type spectral statistic and the usual linear spectral statistic of sample covariance matrices.

翻译：在数据维度与样本量同步增长的情况下，我们证明了样本协方差矩阵的线性谱统计量与通过删除该矩阵某列及对应行所得矩阵的线性谱统计量之差的中心极限定理。与以往研究不同，本工作既不要求总体协方差矩阵为对角阵，亦不要求所有阶矩存在。我们的证明方法对现有策略进行了精妙改进，以应对确定两类特征值统计量之差的均值与协方差结构所带来的挑战。此外，我们还建立了此类差型谱统计量与样本协方差矩阵通常线性谱统计量的渐近独立性。