In this paper, our objective is to present a constraining principle governing the spectral properties of the sample covariance matrix. This principle exhibits harmonious behavior across diverse limiting frameworks, eliminating the need for constraints on the rates of dimension $p$ and sample size $n$, as long as they both tend to infinity. We accomplish this by employing a suitable normalization technique on the original sample covariance matrix. Following this, we establish a harmonic central limit theorem for linear spectral statistics within this expansive framework. This achievement effectively eliminates the necessity for a bounded spectral norm on the population covariance matrix and relaxes constraints on the rates of dimension $p$ and sample size $n$, thereby significantly broadening the applicability of these results in the field of high-dimensional statistics. We illustrate the power of the established results by considering the test for covariance structure under high dimensionality, freeing both $p$ and $n$.

翻译：本文旨在提出一个支配样本协方差矩阵谱性质的约束原则。该原则在不同极限框架下表现出和谐的行为，消除了对维度$p$和样本量$n$速率约束的必要性，只要它们均趋向于无穷大。我们通过对原始样本协方差矩阵采用适当的归一化技术来实现这一点。随后，在此宽泛框架内，我们建立了线性谱统计量的调和中心极限定理。这一成果有效消除了总体协方差矩阵有界谱范数的需求，并放松了对维度$p$和样本量$n$速率的约束，从而显著拓宽了这些结果在高维统计领域的适用性。我们通过考虑高维情形下协方差结构的检验（同时释放$p$和$n$的限制），展示了所建立结果的应用效力。