No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional noncentral Sample Covariance Matrices
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Let _n =1/n(_n + ^1/2_n _n)(_n + ^1/2_n _n)^*, where _n is a p × n matrix with independent standardized random variables, _n is a p × n non-random matrix and _n is a p × p non-random, nonnegative definite Hermitian matrix. The matrix _n is referred to as the information-plus-noise type matrix, where _n contains the information and ^1/2_n _n is the noise matrix with the covariance matrix _n. It is known that, as n →∞, if p/n converges to a positive number, the empirical spectral distribution of _n converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of _n and _n, for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all n sufficiently large.
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