We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix [Formula: see text] and its minor [Formula: see text]. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of [Formula: see text] and [Formula: see text]. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.

中文翻译：

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样本协方差矩阵的线性特征值统计差异的波动

我们证明了样本协方差矩阵[公式：见文本]及其次要[公式：见文本]的线性特征值统计差异的中心极限定理。我们发现这种差异的波动远小于单个线性统计量的波动，这是由于[公式：见文本]和[公式：见文本]的特征值之间的强相关性。我们的结果在 Dumitru 和 Paquette 最近的论文中确定了近似高斯场的空间导数的波动。与 Wigner 矩阵的类似结果不同，对于样本协方差矩阵，波动可能完全消失。