Consider the $p\times p$ matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to $p$, when two samples of sizes $n$ and $T$ from the two populations are available, we construct its corresponding sample version. In the high-dimensional regime where both $n$ and $T$ are proportional to $p$, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded. We derive the convergence in probability of these spiked eigenvalues after scaling, and the central limit theorem for normalized spiked eigenvalues.

考虑一下 $p\times p$矩阵是总体协方差矩阵与另一个总体协方差矩阵的逆积的乘积。假设他们的差异在$p$，当两个样本的大小 $ñ$ 和 $Ť$从这两个总体中，我们构建其相应的样本版本。在高维政权中$ñ$ 和 $Ť$ 与...成正比 $p$，我们研究了当尖峰的数量发散并且这些尖峰不受限制时，样本（尖峰）Fisher矩阵的尖峰（尖峰）特征值的极限定律。我们得出了缩放后这些尖峰特征值的概率收敛性，以及归一化尖峰特征值的中心极限定理。