Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices
نویسنده
چکیده
Let B n = (1/N)T 1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix T n. It is shown in Bai and Silverstein (1998) that, under certain conditions on the eigenvalues of T n , with probability one no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large n. For these n the interval corresponds to one that separates the eigenvalues of T n. The aim of the present paper is to prove exact separation of eigenvalues, that is, with probability one the number of eigenvalues of B n and T n lying on one side of their respective intervals are identical for all large n.
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