Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $ where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix, representing the information, and $ \bbT_{n} $ is a $ p \times p $ non-random nonnegative definite Hermitian matrix. Under some conditions on $ \bbR_n \bbR_n^* $ and $ \bbT_n $, it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ p $ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals.

翻译：设 $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $，其中 $ \bbX_n $ 是 $ p \times n $ 矩阵，其元素为独立标准化随机变量，$ \bbR_n $ 是 $ p \times n $ 非随机矩阵，代表信息，$ \bbT_{n} $ 是 $ p \times n $ 非随机非负定 Hermite 矩阵。在 $ \bbR_n \bbR_n^* $ 和 $ \bbT_n $ 的某些条件下，已证明：对于极限谱分布支撑集外的任意闭区间，当所有 $ p $ 充分大时，几乎必然没有特征值落入该区间。本文旨在继续研究极限谱分布的支撑集，并证明存在精确分离现象：几乎必然地，适当数量的特征值位于这些区间的两侧。