Under the high-dimensional setting that data dimension and sample size tend to infinity proportionally, we derive the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix. Different from existing literature, our results do not require the assumption that the population covariance matrices are bounded. Moreover, many common kernel functions in the real data such as logarithmic functions and polynomial functions are allowed in this paper. In our model, the number of spiked eigenvalues can be fixed or tend to infinity. One salient feature of the asymptotic mean and covariance in our proposed central limit theorem is that it is related to the divergence order of the population spectral norm.

翻译：在数据维度和样本大小往往成比例无限的高度设置下,我们得出了大维样本共变矩阵线性光谱统计的中央限理论(CLT),与现有文献不同,我们的结果并不要求假设人口共变矩阵是相互结合的。此外,本文允许实际数据中的许多共同内核功能,如对数函数和多元函数。在我们的模式中,加注的光值的数量可以固定或趋向于无限。在我们提议的中心限制中,一个突出的特征是,无色平均值和共变变量与人口光谱规范的差异顺序有关。