Exact separation of eigenvalues of large dimensional sample covariance matrices
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)].
scientific article; zbMATH DE number 1462562