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No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. (English) Zbl 0937.60017
Summary: Let \(B_n=(1/N)T_n^{1/2}X_nX^*_nT_n^{1/2}\), where \(X_n\) is \(n\times N\) with i.i.d. complex standardized entries having finite fourth moment and \(T_n^{1/2}\) is a Hermitian square root of the nonnegative definite Hermitian matrix \(T_n\). It is known that, as \(n\to\infty\), if \(n/N\) converges to a positive number and the empirical distribution of the eigenvalues of \(T_n\) converges to a proper probability distribution, then the empirical distribution of the eigenvalues of \(B_n\) converges a.s. to a nonrandom limit. We prove that, under certain conditions on the eigenvalues of \(T_n\), for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all \(n\) sufficiently large.

MSC:
60F15 Strong limit theorems
62H99 Multivariate analysis
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