In this paper, we extend the CLT for sample spiked eigenvalues in the generalized spiked covariance model proposed in Jiang and Bai (2021a) to the case where RDS is considered free, i.e., except for an upper limit of the RDS to guarantee that the spiked eigenvalue is distant, there is no limit for $p/n$, which is the Ratio of Dimension to sample Size (RDS). Therefore, the choice of dimensionality and sample size is more flexible in our regime.

From our work, the limitation of RDS for the CLT for spiked eigenvalues of the sample covariance matrix under usual high-dimensional settings is removed, and the ultra-high dimensional regime is also covered as a special case.

在本文中，我们将江和白（2021a）提出的广义尖峰协方差模型中样本尖峰特征值的 CLT 扩展到 RDS 被认为是自由的情况，即除了 RDS 的上限以保证尖峰特征值很远，没有限制$p/n$，即维度与样本大小的比率 (RDS)。因此，在我们的制度中，维度和样本大小的选择更加灵活。

从我们的工作中，消除了在通常的高维设置下，CLT 对样本协方差矩阵的尖峰特征值的 RDS 限制，并且超高维状态也作为一种特殊情况进行了介绍。