The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form Sn = 1 nBnXnXn∗B n∗, where Bn is a p × m nonrandom matrix and Xn is an m × n matrix consisting of i.i.d standard complex entries. p/n → c ∈ (0,∞) as n →∞ while m can be arbitrary but no smaller than p. We first prove that under some mild assumptions, with probability 1, for all large n, there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large n. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.

中文翻译：

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一般样本协方差矩阵特征值的一些强收敛定理

本文的目的是研究更一般人群下样本协方差矩阵的光谱特性。我们考虑如下形式的一类矩阵小号n = 1 n乙nXnXn*乙 n*， 在哪里乙n是一个p × 米非随机矩阵和Xn是一个米 × n由 iid 标准复数条目组成的矩阵。p/n → C ∈ (0,∞)作为n →∞尽管米可以是任意的但不小于p. 我们首先证明，在一些温和的假设下，概率为 1，对于所有大n，在所有足够大的极限分布的支持之外的开区间中包含的任何闭区间中都没有特征值n. 然后我们得到极值特征值的强收敛结果作为白音定律的扩展。