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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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