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Exact separation of eigenvalues of large dimensional sample covariance matrices. (English) Zbl 0964.60041
This paper is devoted to the study of the fine structure of the spectra of large random matrices \(B_n={1\over N}X_nX^*_n T_n\), where \(X_n= (X_{ij})\) is an \(n\times N\) matrix consisting of i.i.d. standardized complex random variables, \(T_n\) is an \(n\times n\) nonnegative definite nonrandom matrix; the limit \(n/N\to C>0\), \(n\to\infty\), is considered. It is assumed that the limiting eigenvalue distribution function (e.d.f.) \(F_T\) exists, and the \(T_n\) are bounded. It is known that under certain conditions on the distribution of \(X_{ij}\), the limiting e.d.f. \(F_B\) and \(F_{\underline B}\) exist, where \(\underline B_n={1\over n} X_n T_nX^*_n\). It is proved that if the \(T_n\) are such that there are eigenvalues \(\lambda^{(T_n)}_{i_n}\) and \(\lambda^{(T_{n+1})}_{i_{n+1}}\) situated at different sides of an interval \(Y\), then the same is true with probability \(1\) for the eigenvalues of \(B_n\), in the limit \(n\to\infty\). \(Y\) is determined in terms of \(F_{\underline B}\). This theorem improves the results by the same authors [ibid. 26, No. 1, 316-345 (1998; Zbl 0937.60017)].

MSC:
60F99 Limit theorems in probability theory
15B52 Random matrices (algebraic aspects)
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